1 Corinthians 11:1-17:
Be ye followers of me, as I also am of Christ. Now I praise you, brethren, that in all things you are mindful of me and keep my ordinances as I have delivered them to you. 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 is God. Every man praying or prophesying with his head covered disgraceth his head. But every woman praying or prophesying with her head not covered disgraceth her head: for it is all one as if she were shaven. For if a woman be not covered, let her be shorn. But if it be a shame to a woman to be shorn or made bald, let her cover her head. The man indeed ought not to cover his head: because he is the image and glory of God. But the woman is the glory of the man. For the man is not of the woman: but the woman of the man [c.f. Genesis 2-3]. For the man was not created for the woman: but the woman for the man. Therefore ought the woman to have a power over her head, because of the angels. But yet neither is the man without the woman, nor the woman without the man, in the Lord. For as the woman is of the man, so also is the man by the woman: but all things of God. You yourselves judge. Doth it become a woman to pray unto God uncovered? Doth not even nature itself teach you that a man indeed, if he nourish his hair, it is a shame unto him? But if a woman nourish her hair, it is a glory to her; for her hair is given to her for a covering. But if any man seem to be contentious, we have no such custom, nor the Church of God  Now this I ordain: not praising you, that you come together, not for the better, but for the worse.

According to St. Paul, women veil themselves as a sign that His glory, not ours, should be the focus at worship, and as a sign of our submission to authority. It is an outward sign of our recognizing headship, both of God and man, and a sign of our respecting the presence of the Holy Angels at the Divine Liturgy. In veiling, we reflect the divine invisible order and make it visible. This St. Paul presents clearly as an ordinance, one that is the practice of all the churches.

Some women, influenced by the thoughts of "Christian" feminists, believe that St. Paul was speaking as a man of his time, and that this ordinance no longer applies. The tradition and ordinance of veiling is not a matter of Paul being influenced by his culture or sexism; it is a symbol that is as relevant as the priest's cassock and the nun's habit.

Note, too, that Paul is in no way being "misogynist" here. He assures us that, while woman is made for the glory of the man even as man is made for the glory of God, "yet neither is the man without the woman, nor the woman without the man, in the Lord. For as the woman is of the man, so also is the man by the woman: but all things of God." Men need women, women need men. But we have different roles, each equal in dignity -- and all for the glory of God (and, of course, we are to treat each other absolutely equally in the order of charity!). The veil is a sign of our recognizing these differences in roles.

The veil, too, is a sign of modesty and chastity. In Old Testament times, uncovering a woman's head was seen as a way to humiliate a woman or to punish adulteresses and those women who transgressed the Law (e.g.., Numbers 5:12-18, Isaias 3:16-17, Song of Solomon 5:7). A Hebrew woman wouldn't have dreamed of entering the Temple  without covering her head. This practice is simply carried on by the Church.

 That which is veiled is Holy

Note what Paul says, "But if a woman nourish her hair, it is a glory to her; for her hair is given to her for a covering." We don't veil ourselves because of some "primordial" sense of femine shame; we are covering our glory so that He may be glorified instead. We cover ourselves because we are holy -- and because feminine beauty is incredibly powerful. Consider how the image of "woman" is used to sell everything from shampoo to used cars. Women need to understand the power of the feminine and act accordingly by following the rules of modest attire, including the use of the veil.

By surrendering our glory to the headship of our husbands and to God, we surrender to them in the same way that the Blessed Virgin surrendered herself to the Holy Ghost ("Be it done to me according to Thy will!"); the veil is a sign as powerful -- and beautiful -- as when a man bends on one knee to ask his girl to marry him.

Think of what else was veiled in the Old Testament -- The Holy of Holies!

Hebrews 9:1-8
The former [Old Covenant] indeed had also justifications of divine service and a sanctuary. For there was a tabernacle made the first, wherein were the candlesticks and the table and the setting forth of loaves, which is called the Holy. And after the second veil, the tabernacle which is called the Holy of Holies: Having a golden censer and the ark of the testament covered about on every part with gold, in which was a golden pot that had manna and the rod of Aaron that had blossomed and the tables of the testament. And over it were the cherubims of glory overshadowing the propitiatory: of which it is not needful to speak now particularly. Now these things being thus ordered, into the first tabernacle, the priests indeed always entered, accomplishing the offices of sacrifices. But into the second, the high priest alone, once a year: not without blood, which he offereth for his own and the people's ignorance: The Holy Ghost signifying this: That the way into the Holies was not yet made manifest, whilst the former tabernacle was yet standing.

...The Ark of the Old Covenant was kept in the veiled Holy of Holies. And at Mass, what is kept veiled until the Offertory? The Chalice -- the vessel that holds the Precious Blood! And, between Masses, what is veiled? The Ciborium in the Tabernacle, the vessel which holds the very Body of Christ. These vessels of life are veiled because they are holy!

And who is the All Holy, the Ark of the New Covenant, the Vessel of the True Life? Our Lady -- and by wearing the veil, we imitate her and affirm ourselves as women, as vessels of life.

This one exterior, small act is: