Someone has suggested in his Google docs-posted notes and comments on Biblical Headship that what Paul says in 1 Corinthians 11 about headwear for men and women when they are praying or prophesying relates to the kind of glory their head displays.
Here is how I might translate 1 Cor 11:2-16 per his comments about κεφαλη (kephalê - "head") and δοξα (doxa - "glory") and head coverings:
2 επαινω δε υμας οτι παντα μου μεμνησθε και καθως παρεδωκα υμιν τας παραδοσεις κατεχετε
Now, I praise you because you remember all the things I told you and, just as I passed the traditions on to you, you are keeping them.
3 θελω δε υμας ειδεναι οτι παντος ανδρος η κεφαλη ο χριστος εστιν κεφαλη δε γυναικος ο ανηρ κεφαλη δε του χριστου ο θεος
So I want you to know that the head of every man is Christ, and the head of a woman is her man, and the head of Christ is God.
4 πας ανηρ προσευχομενος η προφητευων κατα κεφαλης εχων καταισχυνει την κεφαλην αυτου
Every man who prays or prophesies with his head covered shames his head, [God].
5 πασα δε γυνη προσευχομενη η προφητευουσα ακατακαλυπτω τη κεφαλη καταισχυνει την κεφαλην αυτης
And every woman who prays or prophesies with her head uncovered shames her head, [her man].
εν γαρ εστιν και το αυτο τη εξυρημενη
For she is one and the same as a woman who's head is shaved.
6 ει γαρ ου κατακαλυπτεται γυνη και κειρασθω
So if a woman does not cover her head, then let her hair be shorn.
ει δε αισχρον γυναικι το κειρασθαι η ξυρασθαι κατακαλυπτεσθω
But if it's shameful for a woman to have her hair shorn or to have her head shaved, then let her cover her head.
7 ανηρ μεν γαρ ουκ οφειλει κατακαλυπτεσθαι την κεφαλην εικων και δοξα θεου υπαρχων
A man should not cover his head since he is the image and glory of God [, who is his head].
η γυνη δε δοξα ανδρος εστιν
But a woman is the glory of [her] man.
8 ου γαρ εστιν ανηρ εκ γυναικος αλλα γυνη εξ ανδρος
For man did not come from woman, but woman came from man.
9 και γαρ ουκ εκτισθη ανηρ δια την γυναικα αλλα γυνη δια τον ανδρα
Indeed, man was not created for the woman, but woman was created for the man.
10 δια τουτο οφειλει η γυνη εξουσιαν εχειν επι της κεφαλης δια τους αγγελους
For this reason the woman should have [a sign of]* authority on her head, because of the angels.
* While many (including myself at one time) argue that because "a sign (or symbol) of" isn't present in the Greek text it is therefore an incorrect interpolation for a translation to add it here (as many do), BDAG (Bauer, Danker, Arndt, Gingrich, A Greek-English Lexicon of the New Testament and Other Early Christian Literature, 3rd ed.) on εξουσια says:11 πλην ουτε γυνη χωρις ανδρος ουτε ανηρ χωρις γυναικος εν κυριω
7...It is abstract for concrete, as βασιλεια  [i.e., kingship, royal power] in Diod[orus] S[iculus] [1st Century BC] 1., 47, 5: a stone figure εχουσα τρεις βασιλειας επι της κεφαλης = that wears three symbols of royal power [diadems] on its head.So just as the diadems on the statue are referred to by the abstract term "royal power," and they are said to be on the statue's head, I think the argument can indeed be made that the veil or covering the woman in 1 Corinthians 11 is to wear "on her head" is very possibly being referred to in 11:10 as "a sign or symbol of" authority that she is to wear. Thus, the phrase εξουσιαν...επι της κεφαλης does not necessarily mean that the woman has "authority...over her [own] head." Rather, επι της κεφαλης can (per this statue description) simply mean the place she is to display such a sign or symbol of authority (i.e., "on her head"), and not that over which she has authority.
Knowing nothing more about the statue than what BDAG quotes about it (e.g., I don't know who or what the statue represents), the statue figure might be wearing a sign of its own royal power on its head. Thus, while a woman might be being told here in 1 Corinthians to wear "a sign of" authority on her head, it could be a sign or symbol of HER OWN authority that she possesses for and over herself.
However, woman is not apart from man, nor is man apart from woman in the Lord.
12 ωσπερ γαρ η γυνη εκ του ανδρος ουτως και ο ανηρ δια της γυναικος
For just as the woman came from the man, so also man comes from woman.
τα δε παντα εκ του θεου
And all things come from God.
13 εν υμιν αυτοις κρινατε πρεπον εστιν γυναικα ακατακαλυπτον τω θεω προσευχεσθαι
You judge for yourselves: Is it fitting for a woman to pray to God with her head uncovered?
14 ουδε η φυσις αυτη διδασκει υμας οτι ανηρ μεν εαν κομα ατιμια αυτω εστιν
Doesn't the nature of things itself show you that if a man has long hair, it is dishonorable to him?
15 γυνη δε εαν κομα δοξα αυτη εστιν οτι η κομη αντι περιβολαιου δεδοται αυτη
But if a woman has long hair, it is related to her glory – because she has been given long hair as a covering for her head?
16 ει δε τις δοκει φιλονεικος ειναι ημεις τοιαυτην συνηθειαν ουκ εχομεν ουδε αι εκκλησιαι του θεου
Now, if someone wants to be contentious about this, I want them to know that we do not have a custom like this, nor do the churches of God.
Some Final Comments
As I stated, one reason I interpreted and translated εξουσιαν εχειν επι της κεφαλης in 1 Corinthians 11:10 the way I did – i.e., "should have [a sign of] authority on her head" – was because my translation of the passage was based on how I would do it in light of the statements made in the writer's notes and comments about κεφαλη and δοξα and head coverings. Per the example in BDAG, I concluded that this translation of εξουσιαν εχειν επι της κεφαλης, in line with the writer's thoughts on this matter, can probably be supported. What I cannot determine, though, is whether such a sign of authority represents someone else's authority over the woman or the woman's own authority over her own head. Furthermore, if εξουσιαν εχειν επι της κεφαλης simply means to have one's own authority over one's own head or self, then the translation of this phrase as "[a sign of] authority" is incorrect.
As a side note, some argue that κατα κεφαλης εχων (lit. "downward-from head having") in 11:4 is referring to (long) hair, and not to an additional head covering, and that the adjective in 11:5,13 from ακατακαλυπτος (its component parts can literally mean "not-downward-covered/hidden") is referring to the absence of long hair. The verbs in 11:6-7 from κατακαλυπτω have the literal component parts meaning "to downward-from cover/hide," and also don't specifically refer to a covering other than one's own hair. Such an interpretation perhaps explains the comments about κομαω ("to wear-long-hair/let-hair-grow") in 11:14-15. If this understanding is correct (though it is not without its problems, and no version I know of translates the passage this way), then the debate on whether or not women should cover their heads in church is based on a misunderstanding and mistranslation of this passage.
- I can't clearly tell which parts of what Paul wrote here are from himself, or are his quoting of what someone else said or wrote to him, or are his response(s) to what someone else said or asked.
- When Paul says in 11:16 that neither his group nor the churches of God have such a custom as this, I can't tell if he's referring to the custom of women not being covered when praying or prophesying, or the custom of women being covered when praying or prophesying.
- If, as Paul writes in 11:15, nature has given woman long hair in place of a "covering," and if a woman is to cover her head when she is in God’s presence, wouldn't that support a long-haired woman NOT having to cover her head when praying or prophesying? I'm not sure that one can assume or conclude that γυνη δε εαν κομα δοξα αυτη εστιν in 11:15 means that a woman's long hair "displays" her glory and hence argue that her uncovered head competes with God's glory, for Paul seems to be saying that the long hair acts as a covering for the woman - and hence "covers" her glory. The meaning of this verse is unclear.
- As explained above, I can find support for translating 1 Corinthians 11:10 as "[a sign of] authority on her head." But it could be a sign of her own authority over her own head, and not a sign of someone else's authority over her.
- I still am unable to determine what Paul is finally concluding or affirming or rejecting here, or what he is telling them to do or not do.
- I think #5 would also be the case for just about everyone else who wrestles with this passage.
- The confusing nature of the passage and the different cultural assumptions and practices (along with a nearly two-millennia time gap) that separate us from the writer and his audience make it nearly impossible for me to conclude, as some do, that this passage teaches that men today are to pray or prophesy with their heads uncovered and women today are to pray or prophesy with their heads covered, or that a woman is to have a man (father, husband, pastor) as her spiritual "covering."
- And as I note in my Final Comments, this passage may not even be about head coverings, but about long hair.