This is the strongs word:
Quote:G2619
κατακαλύπτω
katakaluptō
kat-ak-al-oop'-to
From G2596 and G2572; to cover wholly, that is, veil: - cover, hide.
So, I wouldn't say that it could mean "veiling your headcovering". Though, since the whole passage seems to be figurative, I think that "hide" would fit better (though "cover" fits well with the picture that he is painting).
Paul tells you right off:
Quote:1Co 11:3 And I wish you to know that the head of every man is the Messiah, and the head of woman is the man, and the head of Messiah is Elohim.
So, whenever the "head" is mentioned, look here to see what he is talking about. Also, notice that the word for man here can mean husband, and the word for woman can mean wife.
Quote:G435
ἀνήρ
anēr
an'-ayr
A primary word (compare G444); a man (properly as an individual male): - fellow, husband, man, sir.
G1135
γυνή
gunē
goo-nay'
Probably from the base of G1096; a woman; specifically a wife: - wife, woman.
Quote:1Co 11:4 Every man praying or prophesying, having his head [Messiah, though looking at the strong's numbers, this word does not appear to be in the greek to me] covered [hidden], brings shame to his Head [Elohim].
So, if the man hides the Messiah (keeps him out of mind?) while praying, it is dishonoring to Yahuweh, and if the woman does not hide the man (stop thinking about him?) while praying, she is dishonoring Yahuweh.
I'm sure I'm streching here, but I can't imagine that this passage is telling us that men can not to wear hats and women have to wear hats or veils.