I’m not sure what the verse means, but the context of 1 Corinthians 11:1-5 appears to be authority, something the angels would be familiar with.
“……by the resurrection of Jesus Christ: Who is gone into heaven, and is on the right hand of God; angels and authorities and powers being made subject unto him. (1 Peter 3:21-22)
And angels are looking into our affairs.
“Searching what, or what manner of time the Spirit of Christ which was in them did signify, when it testified beforehand the sufferings of Christ, and the glory that should follow. Unto whom it was revealed, that not unto themselves, but unto us they did minister the things, which are now reported unto you by them that have preached the gospel unto you with the Holy Ghost sent down from heaven; which things the angels desire to look into.” (1 Peter 1:11-12)