Later: The author of the blog mentioned above, Joseph Herrin, also has a website where he discusses many points of theology. I was pleased to discover that he spends quite a bit of time on the question of women's headcovering, making some points about God's government I hadn't fully appreciated until now. This is the basic argument, and this is where he gets into the meaning of God's government as opposed to God's grace in more detail.
Something I had run across in my study of the headcovering was the statement that we are to understand that Christ's head is covered before the Father, and the man's head is covered before Christ, just as the woman's head is to be covered before the man. Since the man and Christ do not literally cover their heads -- and in fact Paul says just as emphatically that the man should not cover his head as that the woman should cover hers -- I didn't understand what this meant, but Joseph Herrin has made it quite clear: the woman does it as representative of both the man before Christ and Christ before the Father.
When a sister covers her head, she is standing before God on the basis of Christ’s position before God and man’s position before Christ. God wants the woman to cover her head in order to manifest His government on earth. This privilege falls only to woman. She does not cover her head merely for her own self; she does it representatively. For her own self, she does it because she is a woman; representatively, it is because she represents man before Christ and Christ before God. So when woman covers her head before God, it is just the same as if Christ covered His head before God. Likewise, when woman covers her head before man, it is just the same as if man covered his head before Christ. Man or woman should have no head because Christ is head. If one’s head is not covered, there will be two heads. Between God and Christ one head must be covered; so too must it be between man and woman, and so between Christ and every man. If one head is not covered, the result will be that there are two heads. If God is head, then Christ is not; if Christ is head, then man is not; and if man is head, then woman is not.
I also appreciate the clarity he gives here to the meaning of the symbolism of the head covering, and it makes perfect sense out of all the confusion about it: the head is covered as if to say, I do not have a head. There can only be one head, so the subordinate has to yield, signifying it by covering the head.
One thing he doesn't answer clearly, it seems to me, is whether the woman is to cover her head outside of church as well as in prayer and prophesying, though it seems to be implied that it would be fitting in all circumstances, just as it was understood to be for the last couple millennia until the 20th century.
Later: He does answer it here. Since scripture isn't specific about it he doesn't have a conclusion of his own, but leaves it to the women. His own wife and daughter have chosen to cover their heads at all times. I've been covering mine only when I know I'm going to be praying, but that's problematic because I often stop to pray when I don't expect to be doing it. So I think like his wife and daughter I should wear a covering all the time too.