GOD'S glory is to be seen alone in the assembly of the saints. In order to do this, the men remain uncovered by not having long hair and by removing any head coverings, because the man is "the image and glory of GOD" (1 Cor. 11:7). Any covering on the man would veil GOD'S glory. The women, however, are the stewards of the coverings.
There are two competing glories in the church. "The woman is the glory of the man" (1 Cor. 11:7) and "if a woman have long hair, it is a glory to her" (1 Cor. 11:15). Because there are two symbolic glories to be covered, there must be two coverings. The first covering is the woman's long hair (v. 15b) to hide the glory of the man (the woman herself). The second covering is to hide her glory — her own hair. In this way, GOD'S authority is declared in the church. By it the men are reminded that, in their ministry, their glory is to be hidden. The angels are also instructed by it (1 Cor. 11:10).