The instruction Paul gave to the Corinthians had nothing to do with literal head coverings
Q. I see men praying with their heads covered. Is this biblical?
A. There is a lot of confusion about this practice so I am going to give what I believe about it, not only concerning men but women, too. The best place to learn about this will be 1 Cor 11.1-16 and I want to go verse by verse so that we can get the whole context. You have seen Jewish men wearing a "kippah" or yarmulke and many teach that this is a biblical thing to do.
But in reality there is no commandment for men to wear a head-covering. A priest was to cover his head while ministering in the Temple but that was because he was anointed with the holy anointing oil, but that certainly did not apply to a non-priest.
The practice of covering the head is one of those traditions which evolved over the centuries. Even the movie "Fiddler on the Roof" comments on the head-covering tradition and when Tevye asks about its origin he says "I don’t know" and we all have heard the song "Tradition" from the movie. The kippah/yarmulke shape is interesting. Some say it relates back to the practice of the idolaters to shave or "tonsure" the hair on the head in a circle because they were sun worshippers.
Some denominations have a history of pagan sun worship and they wear head-coverings in the shape of a sun disk and the modern kippah is patterned after that. But, like Tevye, they’d have to say when asked about its origin "I don’t know" so let’s look at what Paul says and let’s put it into a first-century biblical, Jewish context because that is where he was coming from.
In Chapter 11, he begins to comment about the proper relationship between a husband and a wife in public worship in regards to authority. Being a Greek city these Gentile believers were heavily influenced by its culture and traditions.
Paul is trying to teach the Corinthians about the Torah and how to walk in the scri ptures which was totally different to the culture and thought processes they were familiar with. That’s why he starts out in Chapter 11 by exhorting them to follow him because he follows Yeshua. We know that Yeshua followed the Torah so that means Paul followed the Torah and he tells the Corinthians to follow the Torah.
He praises them because they follow the traditional and biblical concepts he has taught them so far.
He then gives the spiritual application of what he is about to teach them in verse 3 and it is this verse that is the basis on how v. 4-16 should be interpreted. He says Messiah is the head of the man and the man is the head of his wife and God is the head over Messiah.
What he is basically saying is he, Paul, is not the "head." He has already clarified that fact in 1 Cor 1.11-17. So he gives a basic spiritual hierarchy and then begins to deal with a problem in the congregations there.
In verse 4, he says any man who has his head "covered" (by any other man or institution) disgraces his head, who is Yeshua. We are not to follow or submit to any man-made authority which violates what the Lord has said (Acts 5.29).
In verse 5, he goes on to say that any woman (wife) who has her head un-covered (by ignoring her husband’s authority as specified in v 3) disgraces her head (husband) and she is like one who "shaves her head." He is referring to the practice of prostitutes who shaved their hair off. In other words she is like a spiritual prostitute because she has "cut-off" her spiritual covering. As you can see Paul is teaching Torah concepts on how a husband and a wife relate to each other especially in regards to public worship (praying and prophesying/teaching).
He says that for a woman (wife) to shave her head literally would be disgraceful, so it is also disgraceful for her to "shave her head spiritually" by going against her husband’s authority.
In v7 he says it is disgraceful for a man to have his head covered (by any authority or institution other than Messiah) since he is the image of God and the woman is the glory of her husband. This is because the woman came from man and was created for his sake. Therefore a wife should have her husband as her authority.
He goes on to say that this is done "because of the angels" and he is referring to that fact that some angels weren’t satisfied with their place and tried to usurp authority that was not given to them and they fell.
So it is a warning about going against God-given authority, especially in a family. In v 13 he says we are to judge for ourselves as to whether it is proper for a wife to pray to God by going against her husband (uncovered).
He says even nature itself teaches us if a man has long hair he dishonors himself, but if a woman does it is a glory to herself. In a spiritual sense it is the same way. A husband who covers himself with any man-made institution or authority (long hair) dishonors Messiah but a wife who covers herself with her God-given "covering" (her husband) is doing the right thing because her glory (husband) was given to her, just like her hair in a natural sense. Her hair is a built-in "badge" of her femininity.
Paul concludes this teaching by saying if anyone wants to argue about what he is saying "don’t bother" because there is no other practice in the congregations. So, in short, Paul is not teaching about literal head-coverings. There is no Torah teachings about that. But, there is plenty of teaching in the scri ptures (Gen 2, Num 30 for instance) about how a husband and wife are to relate to each other, especially in a congregational setting and that is what he is talking about. Evidently the Corinthians were having some problems in this area very similar to what we experience in this culture today because our western culture is based on Greek thought as opposed to Hebraic thought and few husbands and wives today are taught God’s ways because we are not taught the Torah. So, it was the same back then. These people were coming out of a Torah-less society and Paul had to teach them the "traditions" (which in Greek is the word " Paradosis") which basically means "tradition by instruction based on the Torah." These principles hold true for today also. A man is not to have any spiritual institution or authority over him other than the Messiah/Word of God and his wife should place herself under his authority and not rebel against his leadership in a spiritual or congregational setting. Now if she believes he is wrong, she can submit him to his head (Messiah) in prayer and let God deal with him. David did the same thing to Saul when he said he would let God judge between them, but he (David) was not going to lift his hand against him because he was his (David’s) authority as King and placed there by God. This brings up all sorts of questions and scenarios, but that is the basic instruction Paul is giving to the Corinthians and it has nothing to do with literal head-coverings.