Paul specifically wrote that his basis for requiring head coverings is the order of creation which is timeless and universal in scope. The head/skull of the man who is praying/prophesying requires that it be uncovered since his head reflects the image/glory of God (v.7). Therefore to cover his head would be to cover up the image and glory of God - which should always be paramount and made manifest. God's glory is never to be put under a cover. However, women are to cover their heads since the woman is the glory of man. When the church composed of men and women gather to pray and prophesy, God's glory is made manifest when the men leave their heads uncovered while the women do the opposite and cover their head since the woman reflects the glory of man (v.7). In the hierarchical scheme of creation, man's glory is always subordinate to God's glory when the church gathers to pray and prophesy. This also serves as a witness and reminder to the angels (v.10). For the woman however, her long hair is her glory (v.15). That is why she is to wear a head covering since her glory must be covered up. V.15 is a contentious verse as some interpret this verse as the basis for claiming that a woman's long hair is her covering. The verse reads:
"But if a woman have long hair, it is a glory to her, for her hair is given her for a covering."
The word "for" in this verse is from the Greek "anti' which means in exchange for or instead of, in place of. Thus I believe the verse should read:
"But if a woman have long hair, it is a glory to her, for her hair is given her instead of a covering."
Again, since long hair is a woman's glory, she is to wear a head covering to cover up her glory while praying/prophesying in the congregation. The man on the other hand, his head remains uncovered while praying/prophesying in the congregation since his head reflects the glory of God.