Religious context: Romans covered heads during worship, prayer, prophecy, and sacrifice

Religiously, when Romans performed worship activities (sacrifice, prayer, prophecy), both men and women covered their heads. This is widely agreed upon but neglected in many commentaries. Paul's concern is specifically 'when praying or prophesying' (vv.4, 5, 13). This religious context is the key to understanding 1 Corinthians 11 -- you don't even need to resolve the question of general public head covering practices.