The Idea in Brief
The passage at hand is not speaking about hats or headgear worn on the head, but instead speaks to relative hair length, which is beauty for the female; for the male therefore the shorter hair represents the proper exercise of the authority (especially as concerns the communication of the Word of God). To flip-flop this arrangement between men and women is to incur shame.
God's glory is his authority and power, which the Psalms mention at extensive length. When the authority of God appears in man, the head is anointed. One significant passage in this regard is the anointing of the Christ, who represents the absolute authority of God.
Acts 10:38 (NASB)
38 You know of Jesus of Nazareth, how God anointed Him with the Holy Spirit and with power, and how He went about doing good and healing all who were oppressed by the devil, for God was with Him.
The head represents the power. The Spirit (in the form of a dove) descended on his head (Mark 1:10). Thus as the anointed king he is the "head" of state, whose crown is his glory (which represents heavenly authority from God). While he may wear crowns in some literal sense, his "crown" of authority came from the anointing on his head.
In the Christian New Testament, such power comes with the spiritual gifts of "prophesying." The following graph summarizes the general Christian New Testament concept of "prophesying."
In the passage at hand in 1 Corinthians 11:1-22, Paul is referring to women who "prophesy and pray," which are the exercise of the office of speaking gifts.
1 Corinthians 11:4-5 (NASB)
4 Every man who has something on his head while praying or prophesying disgraces his head. 5 But every woman who has her head uncovered while praying or prophesying disgraces her head, for she is one and the same as the woman whose head is shaved.
Paul is saying that women who exercise authority through speaking gifts are in violation of the divine order. That is, they are not reflecting the glory of man (beauty), but the glory of God (authority). Based on the context of 1 Corinthians 11:1-22, the following graph depicts how the divine order relates to the authority of God (crown of glory) and the beauty of man (crown of glory).
NOTE: The male with long hair would appear in the second row.
According to this graph, the male with shorter hair relative to females exercises the authority of God through "prophesying" -- his glory stems from the "crown" of the authority of God. The woman's glory, on the other hand, is her beauty, and therefore her head is "crowned" (with hair). Her head is covered with hair because she is not exercising the authority of God -- that is, the woman is not "anointed" with God's authority and therefore does not wear the "crown" of authority.
Again, the passage is not about hats (or headgear), but about hair and how hair length is the "crown" of authority in men (and beauty in women).
For example, if a woman would prefer to represent the authority of God (public exercise of speaking gifts), then her head should be uncovered (or shaven) not just so she can appear like a man, but so that she can be in alignment with her actual shame as a woman.
BUT, if a woman just so happens to have short hair (for WHATEVER reasons), then she should just wear a wig. In such a way her "crown" of glory is beauty. The wig is therefore the so-called covering to which Paul refers for such women. Hats and headgear (such as literal crowns) are not in view.
To recapitulate, if the woman has long hair, but tries to represent the authority of God (public exercise of speaking gifts) she is no different than the shameful woman with her head shaven. In such a case where this woman has long hair, she should not have any "crown" of glory (hair) since her "crown" of glory is to exercise the authority of God, for which anointing occurred on an uncovered head (for which the king would wear the crown of authority). Again, the issue at hand is NOT about hats worn on the head, but about authority as reflected in what is beautiful in women ("crown" of hair) and what is beautiful in men ("crown" of authority).
So Paul is using fine nuance - i.e., hair length in tandem with "glory" (for the woman is her "crown" of hair, and for the male his "crown" of authority). In this regard, Paul mentions that men with long hair are "contrary to nature" (1 Cor 11:14)-- that is, the place of long hair is specific only to the woman, whose "glory" is their crown of beauty. It is therefore inappropriate for any man to pray or prophesy (public exercise of spiritual speaking gifts) with hair longer than women.
Finally, when Paul mentions "because of the angels" he is saying that what he is stating is timeless. That is, the long hair / short hair debate is not specific to the First Century of Christianity in the Roman World. When he says "because of the angels" he is saying that the angels recognize the spiritual gifts of speaking through the male, and that principle crosses all cultures and all times in the entire history of the world. Paul is therefore not promulgating patriarchy. That is, the magnificence (glory of the woman) goes beyond the visual externalities and includes "inward beauty" (2 Pet 3:3-4). Finally, Paul reminds his readers that men and women originate one from the other (another timeless fact), and therefore are one flesh and AT PARITY as fellow human beings.
1 Corinthians 11:11-12 (NASB)
11 However, in the Lord, neither is woman independent of man, nor is man independent of woman. 12 For as the woman originates from the man, so also the man has his birth through the woman; and all things originate from God.
Nonetheless, the male (and not the female) in the public exercise of speaking gifts represents the authority of God, and this principle is most relevant in "prophesying" (teaching, correction, and exhortation) of the Word of God, when the speaker exercises the authority and power of God over listeners.
The "Christ" is anointed on the head with the Holy Spirit. He is the "head" of state. That is, he is the head of the body of Christ through the male.
1 Corinthians 11:3 (NASB)
3 But I want you to understand that Christ is the head of every man, and the man is the head of a woman, and God is the head of Christ.
Thus the "glory" of the male is to exercise the office of the teaching and speaking gifts for common edification; that is, men who have the gifts and calling are to exercise the authority of God. Women who do this should have their "crown" removed, which is their hair, since the woman exercising divine authority is contrary to the divine order of creation. Such a woman who aligns herself with men should have her head shaven; on the other hand, any woman with no hair (for whatever reason) should seek to wear a wig, so that she will wear her "crown" of beauty, which is her glory.
Finally, men with long hair are covering their "crown," which is authority bestowed by God. That is, they are covering the glory of function (authority) with the glory of form (beauty). Paul alludes on the argument of nature in this regard (1 Cor 11:14).