Hair and Veils
I've read somewhere a bit of speculation that Paul's comments in 1 Corinthians 11 about hair and coverings were related to the Greco-Roman medical mythology of the time. That's patently silly. Paul was referencing the Old Testament, and the Hebrews had no such notions about hair interacting with semen.
It's really not that complex, though I admit that the Old Testament barely mentions hair styling. Rather, it was established custom across the Levant that women who were married wore some kind of covering on their heads in public. The only females who went uncovered were not yet married. The scarf/veil/whatever was simply a marker that she was taken. Grab her and you could die. Grab an unmarried girl and you would be forced to marry her, and her family would watch closely to make sure you treated her well. Every Semitic nation (there were many) understood that from ancient times.
Granted, it wasn't quite that rigorous in places like Corinth with mixed cultures under Roman government. Still, it's critical during Christian worship. Paul insists that men wear no headgear, but women must wear a symbolic covering "because of the angels".
In our fallen state, we cannot appear before God without a covering for the sake of our sinful nature. The imagery is broadly understood across the Ancient Near East, though there were specific variations. The basic principle is that only highest ranking servants could turn their face toward the ruler; everyone else must be modest. There were various means of showing this; most common was turning the face and eyes downward, putting hands or veil across the face, etc. This was the default until the ruler declared one elevated enough to see God.
Moses was warned he could not see God or his fleshly nature would die. Divine glory is beyond what fallen flesh can bear. Feudal status is everything in this imagery.
The angels are described as covering themselves in the Presence of God because they are under an intervening authority, same as women. There would have been something we would call archangels, but it's not that simple. We don't have a good way of putting this into words, and legalism won't get it. It's a mysterious thing to explain, but not to obey, rather like the prohibition against consuming blood. God places men directly under Christ, but He made women to be under male authority. No further explanation is coming. That is, aside from Paul's comment elsewhere about how Eve was genuinely deceived. Adam instinctively knew better; he was just being stupid.
The business of hair is also not too hard to understand. In Deuteronomy 22:5 we get our biggest clue: There must always be an obvious difference between male and female. Gender bending was an abomination. The business of trying to look like the opposition sex, along with homosexuality, was one of those awful things the elohim opposition taught humanity to spite Jehovah.
Further, throughout the Ancient Near East, women with short hair had rejected their feminine roles and proper social stability. In Israel, if a woman was caught playing the harlot while married, she was executed. If she was caught in that role without marriage, her head was ritually shaved and she was marched through public to shame her. Her glory was taken away.
However, in the Old Testament, men's hair was more complicated. The Priests were required to trim their hair, neither shaving nor growing it long. The prohibition against shaving the natural beard shape and hairline had to do with not mimicking the pagan priests of Egypt. We are aware that men under the Nazirite vow didn't trim their hair for the duration of the vow. Samson was born under it and wasn't allowed to end it, so he could never cut it at all. There were apparently other customary rituals alongside this where some vow ended with head-shaving.
Paul doesn't refer to any of that in his letter to Corinth. Rather, he refers to men who grow out their hair like women. That typically marked homosexual prostitutes in Corinth, along with other parts of the Mediterranean, as part of an effort to appear as effeminate as possible. It was relatively common in Greece, in particular, and remained an abomination to God.
In America today, none of this means much. However, as part of our Covenant in Christ, I suggest that women who attend public worship, or any meeting that constitutes formal worship, should wear something like a scarf "because of the angels" and men must worship without headgear of any kind.