A lot of Paul's statements don't really make sense to me, as they seem to contradict a lot of other things. Then again, they may just be taken out of context.
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. 4 Every man who prays or prophesies with his head covered dishonors his head. 5 And every woman who prays or prophesies with her head uncovered dishonors her head--it is just as though her head were shaved. 6 If a woman does not cover her head, she should have her hair cut off; and if it is a disgrace for a woman to have her hair cut or shaved off, she should cover her head. 7 A man ought not to cover his head, since he is the image and glory of God; but the woman is the glory of man. 8 For man did not come from woman, but woman from man; 9 neither was man created for woman, but woman for man. 10 For this reason, and because of the angels, the woman ought to have a sign of authority on her head.
1CO 11:11 In the Lord, however, 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. 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. 16 If anyone wants to be contentious about this, we have no other practice--nor do the churches of God.
He is making a number of arguments regarding the decorum in their services at the time. The main issue is head coverings, not merely long hair. The hair portion is an illustration that backs up his point. He is contending that women should cover their heads during the services.
a. starting in vs. 3- appeal to the story of the fall. He doubtless had in mind the account of the fall in which it said "her desire shall be fore her husband and he shall RULE over her). To Paul this was no cultural issue. It was a God given directive. He is appealing to the established scripture of his day.
b.Starting in vs 8, appeal to the initial creation order and roles. Woman was a helper for Adam. After sin there was strife among the woman and man, therefore there needs to be a definite authority, and one who is accountable to God for the family as a whole. As to angels, they appear to be upset over the lack of decorum.
c. Starting in vs 11, A concession that they are interdependent...but that all comes from GOD...ie, who are you to argue with what God did? Do you not think he anticipated that they might give all kinds of reasons even in their own time? He is being careful to make appeals to established authorities that are free of connotations of their day.
d. Starting in vs. 13, an appeal to the natural order. (Now some might note that while he says the nature of things, that he has in mind the cultural idea that women have long hair. Obviously it is possible for men to have long hair in nature. But this essentially cultural argument is significant in that he puts it third in the line of reasons, and clearly states that he is putting it in their terms....think for yourself...is it really done this way? It is not his main argument, but one that he is using as a reinforcement to his argument.
e. Starting in verse 16, appeal to authority. Finally he ends with an appeal to ecclesiastical authority. And he also assumes that there would be contention over it. Why? Well, I think that is obvious, even then it seemed like a controversial issue. Here he says it is this way in all the churches of God.
Paul is here basing this teaching off of his understanding of the biblical council regarding roles in the family. These he sees outlined in Genesis, and he gives more details of his view in Ephesians 5. He indicates that the wife submits to the husband. But the husband submits to God. So this is not free rein there. And more than that the husband is to love the wife as Christ loves the church. So this is sacrificial love. And it is love that is directed by God. Essentially the man has a greater responsibility to look out for the interests of all in the family. He is directly accountable to God for them.
Back to the passage in Corinthians...His main stress in the argument is that the head covering is a sign of authority for the woman.
Now, what does it all mean then? Quite possibly there were those who were not covering their head, which revealed a lack of respect, and perhaps indicated that they were ignoring the roles given in the family structure.
It has also been suggested that loose decorum in this regard could associate them with the pagan priestesses of the fertility cults, who were essentially cultic prostitutes.