The chiasm: The best part is in the middle
In the world of Paul, people sometimes used a form of reasoning called a chiasm, where they put the most important part in the middle, with the other points sandwiched around it so that the first point is related to the last, the second to the second last, etc. 1 Cor. 11 contains such a chiasm.
A. Every woman who prays or prophesies with her head uncovered dishonors her head … she might as well have her hair cut off; but if it is a disgrace … then she should cover (5-6)
. . . . . . B. A man ought not to cover his head, since he is the image and glory of God … (:7)
. . . . . . . . . . . . C. man did not come from woman, but woman from man; neither was man created for woman… (:8-9)
. . . . . . . . . . . . . . . . . . X. for this reason a woman ought to have authority upon her own head, because of the angels (:10)
. . . . . . . . . . . . C2. in the Lord woman is not independent of man, nor is man independent of woman. For as woman came from man, so also man is born of woman. But everything comes from God (:11-12)
. . . . . . . . Judge for yourselves: Is it proper for a woman to pray to God with her head uncovered? (:13, could be seen as a repetition of the main point or as part of C2.)
. . . . . . B2 Does not the very nature of things teach you that if a man has long hair, it is a disgrace to him; (:14)
A2 if a woman has long hair …glory … as a covering (:15)
The main point of the passage is that women should have authority over their own heads. That is repeated as “judge for yourselves.”
The Greek word we translate as angels has two meanings: Angels and messengers. Some say “because of the angels” mean “because of the messengers” Human messengers/ spies looked what the church was doing. In that case, it is better for a woman to dress as her family or culture expects, so she and the church will not get a bad name.
Others think it refers back to 1 Cor. 6:3 earlier in the letter – if we will judge angels, we can certainly judge trivial matters. (This possibility sounds more likely to me, as both texts mention judging and angels in close proximity.)
Is this the sole application?
It seems that vs. 2-9 has no application among the theory, and the first – and only – application of it is :10: Women should have authority because of the angels. It is the first “for this reason” in the passage. Yet the majority of everything I ever heard on vs. 3 (man as head), or hair coverings, or woman as the glory of man, are applications not made by scripture itself but by human sexism! Somehow, perhaps, vs. 2-9 should be understood in a way that leads to a woman having authority over her (own) head. Maybe, those who understand it any other way prove they do not understand the text