John Chrysostom
"For this reason the woman ought to have a sign of authority on her head" (1 Cor 11:10). Why so? For all the reasons already stated, and besides, because of the Angels. If you pay no heed to your husband, he says, then be ashamed before the Angels.
Thus the covering is a sign of obedience and subjection. It prompts one to look downward, to be humbled, and to preserve virtue. The virtue and honor of one in subjection consist precisely in remaining obedient.
The man is not commanded to do this, since he is the image of the Master Himself. But the woman is rightly commanded to do it.
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Therefore consider how great a transgression it is when you, who have been granted such authority, dishonor yourself by taking on the appearance of a woman. You do the same as if, after receiving a diadem, you cast it from your head and put on a slave's garment instead.