In verse 2 the apostle praised the church at Corinth for keeping the "ordinance" that he had delivered to them. Let's look at the text in verse 3. "But I would have you know, that the head of every man is Christ; and the head of the woman is the man; and the head of Christ is God." It seems to me that what the apostle is saying is that the head(ship) or authority of every man is Jesus Christ, whom the Bible says "...is Lord of all." (See Acts 10:36) And the head(ship) or authority of the woman is the man and the Head(ship) or authority of the Lord Jesus is God. Verse 3, it seems to me, is the premise of the entire text.
In verses 4-6, the apostle explains the nature of prayer and prophesy. Again the head referenced here really refers back to verse 3. The head of every man, Christ, is dishonored when the man prays or prophesies having his head or cranium covered. The same would also apply to the woman, she dishonors her head, the man, when she prays or prophesies having her head, that is her cranium uncovered. I know this sounds a little confusing but in the context Paul goes on to use a man's short hair, and the woman's long uncut hair to illustrate his point made in verse 3.
In verse 7, Paul gives the reason that a man ought not to cover his head (when he prays or prophesies) because "he is the image and glory of God." Likewise "the woman is the glory of the man." Then in verses 8 and 9, the apostle apparently refers back to the Genesis account of creation. (See Genesis 2:18-22) God did not create man for the woman but rather the woman for the man.
Then in verse 10 Paul wrote, "For this cause ought the woman to have power on her head because of the angels." Reading in context the term, "for this cause..." it would appear that Paul is referring back to his original premise in verse 3, or at least to the previous verses 8-9. Paul did not giving licence to anyone to appropriate to themselves, taking verse 10 out of contest and make a new or different doctrine as to its meaning. Without trying to belittle the King James Version, (it is for the most part the only one I use in preaching, teaching or reading), some other translations render the word "power" in this verse as "authority". The woman's authority derives from her head, the man, per verse 3. If a woman has long uncut hair, it is her glory and she is the glory of the man, not the glory of God which is what Paul said the man was in verse 7.
"Because of the angels..." There are probably as many interpretations of this phrase as there are interpreters. The apostle no doubt knew exactly what he meant. So it seems to me that we have to again go back to the context to understand Paul's meaning. Going back to verse 3, the Bible says that the head, or if you will, the authority the woman has is derived from the man. So assuming the term, "because of the angels," is referring to heavenly beings, it would probably be that they are observing the proper order in the church. (See also 1st Peter 1:11-12)
Paul goes on then to say in verses 11 and 12 that neither the woman or the man are inherently superior to the other but "all things (are) of God." Then in verses 13-15 Paul resumes his reasoning, "Judge in yourselves, is it comely for a woman to pray unto God uncovered?" The answer to this rhetorical question based on the context is no. Then he says, "Doth not nature itself teach you that if a man have long hair, it is a shame unto him? The answer to this rhetorical questions is yes. Paul continues, "but if a woman have long hair, it is a glory to her: (not to God) for her hair is given her for a covering." So the apostle states that the woman's (long uncut) hair is given to her for a covering.
The Apostle Paul caps his discussion by saying that "if any man seem to be contentious, we have no such custom, neither the churches of God." So to my way of thinking, the entire discussion found in 1st Corinthians 11:2-16 is about God's order of things. Christ is the head of every man, the man is the head of the woman and God is the head of Christ as outlined in verse 3. A man who lets his hair grow long and prays or prophesies does dishonor to the Lord Jesus, his head. The woman who cuts her hair and prays or prophesies (although to and for God) dishonors her head, the man. To me, any other explanation is fanciful and non-biblical.