Someone related to me that the covering here is not an article of clothing according to the original language. He explained that the veil that Moses used to cover his face was clearly seen as such in the original language, but we don't see it for the covering in I Corinthians. I have not studied this, yet. Do you have any insight in this?
The articles given on this site do address this issue, but attempt to make it easier to understand by appealing to the wording in the passage while avoiding any complexities in the Greek. See the comments given for verse 5 in Command or Custom? and The Message of I Corinthians 11.
But since you asked about the Greek:
"Every man praying or prophesying, having his head covered, dishonors his head" (I Corinthians 11:4). In Greek this is: kata (on) kephales (head) echon (having); that is, having something on his head.
"But every woman who prays or prophesies with her head uncovered dishonors her head, for that is one and the same as if her head were shaved" (I Corinthians 11:5). In Greek this is: akatakalupto (uncovered) te (with the) kephale (head).
"For if a woman is not covered, let her also be shorn. But if it is shameful for a woman to be shorn or shaved, let her be covered" (I Corinthians 11:6). In Greek, the first phrase is: ou (not) katakaluptetai (covered). The second phrase is a single Greek word: katakaluptestho (let her be covered).
"For a man indeed ought not to cover his head, since he is the image and glory of God; but woman is the glory of man" (I Corinthians 11:7). In the Greek this is: katakaluptesthai (to have covered) ten (the) kephalen (head).
"For this reason the woman ought to have a symbol of authority on her head, because of the angels" (I Corinthians 11:10). In the Greek this is: epi (on) tes (the) kephales (head).
"Judge among yourselves. Is it proper for a woman to pray to God with her head uncovered" (I Corinthians 11:13). In the Greek this is: akatakalupton (uncovered) to (with).
"But if a woman has long hair, it is a glory to her; for her hair is given to her for a covering" (I Corinthians 11:15). In the Greek this is: peribolaiou (of a covering).
These are just the verses that mention a covering in some way. Notice that there are three ways it is mentioned:
- Having something on the head (I Corinthians 11:4, 10).
- Having the head covered, not covered, or uncovered (I Corinthians 11:5, 6, 7, 13)
- Having a covering (I Corinthians 11:15)
In Greek one negates a word by putting a "a" at the beginning, much like putting an "un" in front of words in English. So akatakaluptos is the negative of katakalupto. Greek also has a variety of endings for words to change the number of people referred to, the gender of those people, the confidence of the fact, the time frame it occurred, and other things like this.
So our main concern is the definition of katakalupto. The word means to cover or hide. It is actually a compound word: kata (an intensifier) and kalupto (hide or conceal). It was used in the Septuagint in Numbers 22:5, "See, they cover the face of the earth." Or, "You shall put the mercy seat upon the ark of the Testimony in the Most Holy" (Exodus 26:34). Or, "Above it stood seraphim; each one had six wings: with two he covered his face, with two he covered his feet, and with two he flew" (Isaiah 6:2). In the New Testament, the word is used exclusively in I Corinthians 11, and as pointed out it is something that is put on, or in the negative cases, taken off. It is not the hair itself because if a woman refuses to cover her head, Paul says she should also shave her head.
In I Corinthians 11:15, a different Greek word is used, peribolaiou. It refers to a cloak or garment that is thrown around to put on. It is also used in Hebrews 1:12, "Like a cloak You will fold them up, and they will be changed."
You referenced: "Unlike Moses, who put a veil over his face so that the children of Israel could not look steadily at the end of what was passing away" (II Corinthians 3:13). That word is kaluma which also means to cover, hide, or conceal. It is a different word and really doesn't help explain I Corinthians 11. Notice also that Moses covered his face, but I Corinthians speaks of covering the head.