1Co 11:4 Every man praying or prophesying, having his head covered, dishonors his head.
1Co 11:5 But every woman who prays or prophesies with her head uncovered dishonors her head, for that is one and the same as if her head were shaved.
In Corinthians Paul speaks about men prophesying with their head covered and women prophesying with their head uncovered. Some people interpret these “coverings” as hats. Men cannot wear a hat in church and women must always being wearing a hat.
But that is not the point that Paul seems to be trying to make. Instead the focus is on hair. In verse 5 he talks about being “shaved”. In the following verses he references haircuts even more:
1Co 11:6 For if a woman is not covered, let her also be shorn. But if it is shameful for a woman to be shorn or shaved, let her be covered.
He contrasts “shorn” and “shaven” with “covered”. They seem to be opposites. He goes on a few verses later:
1Co 11:10 For this reason the woman ought to have a symbol of authority on her head, because of the angels…
1Co 11:13 Judge among yourselves. Is it proper for a woman to pray to God with her head uncovered?
1Co 11:14 Does not even nature itself teach you that if a man has long hair, it is a dishonor to him?
1Co 11:15 But if a woman has long hair, it is a glory to her; for her hair is given to her for a covering.
In verse 13 Paul talks about having a head uncovered. In verse 14 he contrasts this to men having long hair. Then in verse 15 he points out that hair is “covering”. The hair is woman’s “glory” (compare this to verse 10).
So in 1 Corinthians, Paul does not seem to be talking about hats and veils, but long and short hair. Men should not have long hair (God does not like it and thus when long haired men try to talk to God or for God, they dishonor themselves). Woman should not have short hair, it is just as if they were bald.