From Haydock’s commentary:
Ver. 3. The head of the woman is the man, &c. To have the head covered at public meetings, is, according to St. Paul, a mark of subjection: The man was created to be head over the woman, who was made subject to the man, being made of him, of his rib, and the woman made for him, not he for the woman. The man in a special manner, is the image of God, not only by his immortal soul, in which sense also the woman was made to God’s image, and likeness, but inasmuch as God gave him a power over all creatures, and so he is called, the glory of God. For these reasons, as well as from a received custom, St. Paul tells every woman, that in prayer or prophesying in public meetings, she must have her head veiled, and covered in testimony of her subjection to man, her head, otherwise she dishonors herself, and her head. This is what he tells her, (ver. 10.) that she ought to have a power over her head, that is, to have a veil or covering, as a mark of man’s power over her: and because of the angels, that is, out of a respect to the angels there present. Some understand the priests and ministers of God, called angels, particularly in the Apocalypse. St. Paul adds, that nature having given to women long hair, designed it to be as a natural veil. In fine, he appeals to them, to be judges, whether it be not unbecoming in women to pray without a veil. But he will have men to be uncovered, and not to bear such a mark of subjection, as a veil is, by which a man would dishonor his head, that is, himself, and Christ, who is his head, and who appointed him, when he created him, to be head over the woman. He looks upon it as a dishonor and a disgrace for men to nourish their hair, as women should do. He also calls God the head of Christ, that is, of Christ, as man. Lest he should seem to lessen the condition of women more than necessary, he adds, that the propagation of mankind now depends on the woman, as well as on the man, seeing every man is by the woman. (Witham)
It’s all symbolic.