Should women wear head coverings?

1 Corinthians 11:10  For this reason, and because of the angels, the woman ought to have a sign of authority on her head.
Now if Paul had written "have eggs for breakfast because of the devil," I would be careful to stock up on eggs, and eat them dutifully every morning. Even if I didn't understand this very well. Sometimes Paul does not explain in detail! As here:
1CO 16:14 Do everything in love.
Just this, and he does not elaborate, or tell us why (or how!) we are to do everything in love, or what the results will be. "Because of the angels" implies to me that this is more than just a dress code, though I don't understand the connection.
Yet certainly 1 Cor. 11 is a difficult passage, with possible cultural references. But we shouldn't give up on that account! And I don't expect the Corinthians understood all that Paul didn't explain, that we find difficult to understand. If so, I want to buy a Corinthian commentary!
1CO 11:15 For long hair is given to her as a covering.
However you interpret this verse, it at least says that long hair for women is from God, and thus this aspect of the discussion is not just a cultural application.
1CO 11:6 ... if it is a disgrace for a woman to have her hair cut or shaved off, she should cover her head.
Now if long hair is given to a woman by God, then I think, yes, it is a disgrace for her hair to be cut off, regardless of the culture. Thus I think "she should cover her head" applies, regardless of culture.
But is a different application possible here? Actually, the question here is not about a different application, but no application at all. No one is suggesting an alternative here, except for the interpretation which says verse 15 means "long hair is in place of a covering." But this makes verse 5 meaningless. If someone has an alternative to a head covering, I would be glad to hear it!
"The phrase (kata kephales echon) is to be interpreted as meaning 'having something on the head' (literally, 'having [something] down from [or over] one's head'), such as a veil (BAG, Kata, I, la)." (Expositor's Bible Commentary) So the grammar itself seems not to fit well with an interpretation of the covering being long hair itself.
But how about hair given "instead of a covering"? The idea of hair being given "instead" of a covering seems unnecessary, since the BAGD Greek lexicon says that the Greek preposition ANTI here can mean "as."
Thus various main points Paul makes are not cultural ones at all, they are theological. To give up head coverings entirely, an explanation is needed as to why the "woman for man" purpose that the covering had is now no longer an effective requirement, or how no covering meets that requirement, and how no covering addresses the angel issue! Or how we know that the angel issue has changed...
Now it also seems that this is in reference to public prayer or speaking in church. So I don't think it's required for all women who attend church. I admire the (mostly elderly) African-American ladies who always wear a hat to church, though they wear a hat when they don't need to, because they don't speak up when they should!
And even if this issue seems a small point, a lot turns on these little points of conscience. "He who is faithful in that which is least"...
Now certainly the head-covering is an application of a principle, apparently of recognizing God's order in creation, among other aspects, but that doesn't mean that we can vary the practice just because it is an application. Caring for the poor is an application, but because we care for other people in another way, doesn't mean we need not do any almsgiving.   Now Paul anticipates debate! So how was this a basic, well-known and accepted cultural principle?
Also, what Paul is saying here is that there is no alternative ("we have no other such practice," v. 16), thus if someone today proposes an alternative to head coverings, this verse is a hurdle that needs to be cleared. And why all this discussion and persuasion, a whole chapter, and what happened to the theological reasons, if it's optional?
Paul also appeals to nature, "does not even nature teach you?" (vv. 14-15) and this is not asking for culture's opinion.
And if Paul's argument is partly a cultural one, that doesn't make it only a cultural application. Paul was trying to persuade people, and apparently, if they wouldn't take the theological reason, then he would appeals to their sense of what is proper. But his first reasons (his real reasons?) the ones that <em>have</em> to be overturned, are the theological ones.
Now some would perhaps make "woman for man" mean "a woman should show submission to man". And then undo that reason by saying that women are not under such structures today. Thus the whole principle disappears. But Paul's principle was based on creation, and is not subject, I think, to such disappearances. And thus I think we should not substitute "no application" for the application Paul gives here.
Also! To address this completely, it needs to be shown why men were told to not cover their heads when they pray. What was the principle, or the cultural application here? Only Paul again refers to a theological principle, "[man] is the image and glory of God." The same problems arise here as in the women having a head covering.
Head coverings for men were used in religious ceremonies, with the priests in those days. So this was then understood in the culture, part of the religious culture, if the priests wore them! Why was it appropriate for the priests to wear them, and not for a layman in Corinth to wear them, in that culture? Men in the temple wore a head covering, during worship, with approval in that culture, so this argues against Paul simply following the cultural norms in this chapter.
If there is not a convincing explanation (r.e. the angels, where any interpretation must be dubious), then I think we had best follow the instruction, and eat what is set before us, and not insist on understanding how "oats, peas, beans and barley grow," before we will be willing to eat them.
As a parallel to consider, baptism used to be a cultural expression of joining a cause. Today it is not. Therefore we need not baptize, because the culture does not use or understand it?
The reasons for baptism may not exist now. Maybe we should have the men shave their heads, like in the army. The culture would understand that better!
If this is taken to be instruction so that the culture would not misunderstand, or be upset, when were the Christians worried about what the culture might think of their worship practices? hen the culture decided that Christians were doing horrors in communion, the Christians kept right on taking communion, even though the culture was scandalized.
Paul does not even examine the motive, that is not in view here. The issue of legalism can been raised, but should people not participate in baptism, because it can be done legalistically?
And what about communion? In Paul's day, the temple sacrifices and the partaking of the sacrifices was a cultural parallel to communion, as was the Passover festival. This cultural expression has now, by and large, disappeared. The culture used to understand sacrifice and partaking of a sacrifice, now in Western culture, by and large, it does not. Should we stop taking communion? Communion was given to remember Christ, and to proclaim the Lord's death until he comes. Are these nevertheless only in reference to the culture, and were they therefore only intended as cultural expressions of remembering and showing the Lord's death?
Why may we not substitute some other ceremony for this, such as a candle-lighting ceremony, or a sunrise service? Or no service at all, if the culture doesn't understand it? The Corinthians started to vary a little bit, to make it a meal or a festival, and Paul said this was much mistaken (the second half of this same chapter, 1 Cor. 11:17-34).
It does bear mentioning, though, that submission is always a prelude to exaltation (Philippians 2!), that men uncovering their heads, and women covering theirs is for a good purpose, God wants to give us something, not to cause us difficulty, but to do us good.

"That Greek women did appear in public without a head covering is evident from ancient Greek vase paintings" (Expositor's Bible Commentary)
I think this whole cultural argument is doubtful! Maybe the Corinthians were resisting because they were <em>not</em> used to this.
And this was not merely advice, to be considered optional:
"In using 'we' [1 Cor. 11:16] (meaning the apostles), Paul teaches that the Corinthians are to take his statements given in the preceding verses as having apostolic authority, and not as pious advice." (Expositor's Bible Commentary)
Now if the principle is "follow the culture when praying", then men who are Jewish believers, and those in Muslim countries, should cover their heads, yet this is the exact opposite of what Paul teaches, which should give us pause.
And if we do not understand these verses well, this should actually make us more prone to carry out the instructions. If you see a warning sign which is partly unreadable, will you not consider taking an alternate route? Instead of saying, well, I don't understand that! Therefore the road must be fine...
"The command was 'take, eat', not 'take, understand'" (C.S. Lewis).
And people can become ill, and even die, on seemingly small points of disobedience, or even for a missing the point of the instruction:
1CO 11:29 For anyone who eats and drinks without recognizing the body of the Lord eats and drinks judgment on himself.
And why are we not eager to do what we even think might be the Lord's will? Is it only possible that these instructions do apply today? Thus why do we not seek to err on the side of doing more, possibly pleasing God, instead of possibly choosing something that would displease him? I don't think anyone would say the Mennonites are sinning when the women cover their heads. I think there is, however, a possibility of sinning if people don't follow the instructions given here, for both men and women.
Back fifty years or so ago, Rees Howells decided that he would not wear a hat any more, because men were not to wear them while praying, and reading the verse "pray without ceasing" in view of this he doffed his hat pemanently. Now I think that Paul meant public prayer, he didn't notice that this implies that women would also have to cover their heads all the time! But I admire him, and those who became "the hatless brigade," and were willing to be misunderstood by their culture, and even went through difficulty, in order not to possibly displease God.
Another factor is that men uncovering the head during prayer seems to continue, even in heaven:
REV 4:10-11 The twenty-four elders fall down before him who sits on the throne, and worship him who lives for ever and ever. They lay their crowns before the throne and say, "You are worthy, our Lord and God..."
But maybe this also brings back footwashing! So why is that not an ordinance? I actually don't mind if it is.
How about "greet one another with a holy kiss"? But there is no theological reason here for this, so I think this is cultural.
Now if we are required to have a verse that head coverings should continue, I cannot produce a verse that says "do this always", but there is no such verse for baptism or communion, either. And if the objection is made that only baptism and communion are ordinances, are only ordinances binding?
1CO 14:37 If anybody thinks he is a prophet or spiritually gifted, let him acknowledge that what I am writing to you is the Lord's command.
So I think the issue of headship, the image and glory of God, purposes of creation, and the angels are all significant reasons, and the glory of God is even paramount, in some theologies. So this is not trying to elevate head covering, rather it's trying to follow these theological reasons.
1CO 11:8-9 For man did not come from woman, but woman from man; neither was man created for woman, but woman for man.
With regard to verse 8, how has the creation of Eve from Adam's rib ever been expressed culturally by a head covering? I think this is solely a theological reason, without any cultural connection. Similarly with verse 9, woman was created for man. I think this refers primarily to companionship, I think "created for man" refers to this purpose for creating woman, before the fall:
GE 2:20-12 But for Adam no suitable helper was found. So the Lord God caused the man to fall into a deep sleep...
Again, how has this ever been expressed in any culture, by the practice of head coverings? So now we have two theological reasons, without any cultural expression. Thus a cultural practice of head coverings will not affect these reasons, since the cultural practices in Paul's day were expressing a different principle.
Also, I don't think the cultural submission of women in Paul's day is in view even in the mention of headship in verse 3, what was taking place in Paul's day, in the social structures which placed men over women, seems to have been a part of the fall:
GE 3:16 "Your desire will be for your husband, and he will rule over you."
But the sense in verse 3, I think, is referring to God's purpose in creation, before the fall, a sense of headship like Christ is head of the church, which is not like any social structures I have heard about.
And why did Paul not instruct women to wear a head covering all through the service, regardless of what they were doing, or when they shopped at the local market, or when they visited the Greek museum? Why only while praying, if the reason was that the culture might mistake an uncovered head for some insubordination?
If women could speak in the public assembly, then why didn't Paul stipulate that a woman who gave out a hymn to sing, or who led in a song, or who sang a solo song of praise, should also cover her head? Surely the culture would notice that as much as praying in public worship! And if women must cover their heads when they prophesy, why shouldn't they cover their heads if they give a public Scripture reading? Or recite a poem?
And if Paul meant "prayer" in the sense of all praying, then he didn't need to mention prophesying, since "pray without ceasing" implied head covering all the time, without the additional mention of prophesying.
If we include more that just praying and prophesying in Paul's instructions, we are now in a mare's nest. All through the church service? When coming to or leaving the service? When meeting informally with friends? When singing hymns in the house? I don't think Paul would have left us so much asea. Rather, I think Paul's instructions do indeed give us the bounds of application, rather than a minimal application, for the act is a symbolic one, and the principles are theological, and the connection with praying and prophesying here I think is exactly what is intended, just as we are to baptize people upon repentance, and not before or afterwards, regardless of whether people were baptized for the dead, or at some other time, or for some other reason. We are given no theological reason to baptize people for the dead, but we are given theological reasons to baptize people upon repentance. So I would proceed similarly with head coverings.
I think we may also say that prayer or prophecy in the church meeting was what was being referred to, because Paul mentions head coverings (or not) as a practice of the churches, implying a practice when people were gathered together. Also, in the discussion starting in verse 17, he continues giving instructions, for which he has no praise for them, in contrast to the instructions starting in verse 1, where he did have praise for them. Thus the instructions seem to be addressing various concerns of a similar nature, and the instructions starting in verse 17 clearly refer to church meetings.
There may even be good grounds for considering head coverings an ordinance, it is a ceremonial act, instituted with divine authority, the reasons given being not the obvious ones, and they are theological reasons.
And how is it that every man should uncover their head, and every woman should cover hers? Not every culture in that day had such a practice, not even every Mediterranean culture! The Greek women apparently went out-of-doors without a head covering, and someone thought it appropriate to paint them in that way. You have to paste in here "every [Roman] woman," or "every [Middle Eastern] man." But that's not what Paul says. Every man or woman, even in his day, which includes people all over the world, in the remotest islands, in the remotest province.
1CO 11:3 Now I want you to realize that the head of every man is Christ, and the head of the woman is man, and the head of Christ is God.
This is not just "every Mediterranean man," here...

Tertullian (born about A.D. 145):
"But that point which is promiscuously observed throughout the churches, whether virgins ought to be veiled or no, must be treated of. For they who allow to virgins immunity from head-covering, appear to rest on this; that the apostle has not defined 'virgins' by name, but 'women'..."
"For indeed it is 'on account of the angels' that he saith women must be veiled, because on account of 'the daughters of men' angels revolted from God. Who then, would contend that 'women' alone — that is, such as were already wedded and had lost their virginity — were the objects of angelic concupiscence, unless 'virgins' are incapable of excelling in beauty and finding lovers? Nay, let us see whether it were not virgins alone whom they lusted after; since Scriptures saith 'the daughters of men;' inasmuch as it might have named 'wives of men,' or 'females,' indifferently."
"Again, while he says that 'nature herself,' which has assigned hair as a tegument and ornament to women, 'teaches that veiling is the duty of females,' has not the same tegument and the same honor of the head been assigned also to virgins?"
"This custom, in short, even Israel observes; but if Israel did not observe it, our Law, amplified and supplemented, would vindicate the addition for itself; let it be excused for imposing the veil on virgins also."
"Why do you denude before God what you cover before men? Will you be more modest in public than in the church?"
"You do well in falsely assuming the married character, if you veil your head; nay, you do not seem to assume it falsely, for you are wedded to Christ: to Him you have surrendered your body; act as becomes your Husband’s discipline. If He bids the brides of others to be veiled, His own, of course, much more."
Bengel (a possible interpretation of "because of the angels"):
"As the angels are in relation to God, so the woman is in relation to man. God’s face is uncovered; angels in His presence are veiled (Isaiah 6:2). Man’s face is uncovered; woman in His presence is to be veiled. For her not to be so, would, by its indecorousness, offend the angels (Matthew 18:10,31)."