My best guess for explaining the passage
I think – but do not teach – that Paul was probably responding to things the Corinthians wrote to him here. (See 1 Cor. 7:1 for evidence Paul was speaking, in the latter part of 1 Corinthians, of some things they wrote to him about.)
He is probably quoting the contradictory views among them, views they probably argued (see “contentious” in vs. 16) about. And then, at the end of stating (mocking?) the way their views contradict each other, Paul gives this message: “Women should have authority on their own heads … Decide for yourself… I say this to anyone who wants to argue.”
In my view, Paul’s message (after quoting the Corinthians) is:
10 It is for this reason that a woman ought to have authority over her own head, because of the angels.
Because all your contradictory arguments come to no conclusion, and believers will judge angels one day, sure women could decide for themselves what to wear on their heads.
11 Nevertheless, in the Lord woman is not independent of man, nor is man independent of woman. 12 For as woman came from man, so also man is born of woman. But everything comes from God.
Despite all your arguments on male source-ship (vs. 3) and women being made for the man, and the glory of man (vs. 7-9) women and men need one another and both have God as a source.
13 Judge for yourselves: Is it proper for a woman to pray to God with her head uncovered? 14 Does not the very nature of things teach you that if a man has long hair, it is a disgrace to him, 15 but that if a woman has long hair, it is her glory? For long hair is given to her as a covering.
Everyone can judge for himself (not for anyone else in the congregation) about (female) headwear/ hairstyles. The nature of things (the culture they are in) gives a clear message of how men should wear their hair. The culture and logic also say long hair is already a covering, women need not worry about further headwear.
(Women’s hair, in that world, was shorn and shaven to shame them, and Paul did not need to tell women anything about choosing short hair – no woman would, in that culture. But, of course, women who were shamed by the culture could “judge for themselves” that they want to cover up their heads before a merciful Christ who do not judge them by past sin.)
16 If anyone wants to be contentious about this, we have no such practice—nor do the churches of God. (Some translations say “no other practice”)
There is no practice in the churches regarding head coverings, and Paul does not want to establish one. Or there is no practice other than letting women decide for themselves, and Paul does not want to establish one. So people should not argue about women’s chosen headwear or hairstyle.
Perhaps my understanding of the passage is incorrect. But I can see no way that all the apparent contradictions of 1 Cor. 11:2-16 could be taken on face value. Certainly, anyone who interprets this from an understanding by which head means leader, or who change a women’s authority to a “symbol of (someone else’s)” authority over her, start from a faulty assumption.
As such, I will humbly admit that I am not sure how Paul reasoned here. But his conclusion is clear: Women should have authority over their own heads. They can judge for themselves what is proper to wear on their heads. The unclear part is the argument Paul used to get to this conclusion.
Neither male nor female: Power over her own head (Remember to click through to page 2 after reading page 1)
Women in the heart of God (9) (See about 1/4 down the page for a discussion on what this passage cannot mean.)