You picked one of the controversial Chapters written by the Apostle Paul.
You can not read just that one verse and expect to reach a full understanding of it in my opinion. You must read the whole Chapter. Thankfully, it isn't one of his longer ones.
Only verses 3 through 16 deal with the subject you are questioning, so it's easier to concentrate on them to see what Paul was saying.
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 isGod.
4Every man praying or prophesying, having his head covered, dishonoureth his head.
5But every woman that prayeth or prophesieth with her head uncovered dishonoureth her head: for that is even all one as if she were shaven.
6For if the woman be not covered, let her also be shorn: but if it be a shame for a woman to be shorn or shaven, let her be covered.
7For a man indeed ought not to cover his head, forasmuch as he is the image and glory of God: but the woman is the glory of the man.
8For the man is not of the woman; but the woman of the man.
9Neither was the man created for the woman; but the woman for the man.
10For this cause ought the woman to have power on her head because of the angels.
11Nevertheless neither is the man without the woman, neither the woman without the man, in the Lord.
12For as the woman is of the man, even so is the man also by the woman; but all things of God.
13Judge in yourselves: is it comely that a woman pray unto God uncovered?
14Doth not even nature itself teach you, that, if a man have long hair, it is a shame unto him?
15But if a woman have long hair, it is a glory to her: for her hair is given her for a covering.
16But if any man seem to be contentious, we have no such custom, neither the churches of God.
In my opinion only I suppose: Paul is discussing the Jewish customs of men and women praying, and whether their heads should be covered or uncovered. I believe (do not know for sure) that this is where the Catholic and some other Churches get their custom of having the woman cover her head when in the Sanctuary. Actually, though, as I read it, the woman only has to have her head covered IF she has hair. If she cuts her hair (shorn), she need not be covered.
Men, on the other hand are to keep their hair "shorn" for it is a shame for him to have long hair. Paul even mentions Nature teaching this, and I have to wonder if he gave this statement a lot of thought prior to writing it. If one looks at Nature, one sees that the male of the species quite often has the longer hair, and not the female. The male is most often the most colorful, and the female the most bland. One clear example is the lion. Male lion has a great mane! Female lion, no. Other examples are common.
In any event, Paul makes it clear he is addressing the "custom" and not declaring a Commandment from God (in my opinion) because he includes the "disclaimer"
[
But if any man seem to be contentious, we have no such custom, neither the churches of God.]
Anyway, the idea of a woman not being allowed to cut her hair, and keeping her head covered probably has some foundation in this passage of Scriptures in my opinion.
It is also worth remembering that the Apostle wrote in two distinct but separate styles....
Thus sayeth the Lord! and It would be better that/I would rather that..............
Recognizing which style he is writing in when reading his Epistles helps to determine if he is relating to the Church Commandments from God, or giving the Church his opinion on how to best live a Christ like life. These writings are not Commandments, but guidelines that will allow a believer to have a more rewarding life in Christ...........in my opinion. If you go back and read Chapter 7, you will see that he clearly differentiates between what are his opinions and what are from God.......
As for me and my church, we agree with verse 16. We, the Church of God Reformation Movement, have no such customs.