1Co 11:7 For a man indeed ought not to cover his head, forasmuch as he is the image and glory of God: but the woman is the glory of the man.We can't join forces with these other groups, and I think it is a mistake to link with them in spirit at all as some Christians seem to be doing. We can acknowledge their rightness by God's standard of male headship, but we can't join with them in any general way. This includes Catholics who also cover or are now learning to cover.
It's important that we keep the gospel testimony pure and separate from all other frames of reference.
I think Christian women who learn to cover now should be seeking a style of covering that sets us apart from these other groups.