I've just been reading through 1 Corinthians again and have seen chapter 11 in a new light.
This is the odd chapter where Paul talks about women being from men and therefore needing to cover their heads. The context of the passage is a longer word from Paul about the freedom for believers from the law (especially the food law). The context includes the verses ""Everything is permissible" - but not everything is beneficial" and ""Everything is permissible" - but not everything is constructive". Here, I believe that Paul is genuinely saying that there is freedom in Christ from the law (to do things which aren't immoral). The caveats are explained by the previous verses: they're about not offending those 'weaker' ones who haven't fully grasped the freedom in Christ. For instance, the weaker one may believe that one is engaging in idol worship by eating idol food - in which case don't do it.
So we come on to 'propriety in worship'. In verses 3-10 of Ch. 11 we're told about conventions in worship in which women should cover their heads and men needn't "since he is the image and glory of God; but the woman is the glory of man. For man did not come from woman, but woman from man; neither was man created for woman but woman for man."
What I've just noticed for the first time is that Paul then contradicts both of these arguments for head covering. I believe that in verses 3-10 Paul may be reflecting the common view (the context of law) whereas in verses 11-16 he explains why this justification doesn't work and that there is freedom.
In this reading, the world's v8: "for man did not come from woman..." becomes Paul's v12: "for as woman came from man, so also man is born of woman."
The worlds' v7: "since he is the image and glory of God: but the woman is the glory of man" becomes Paul's v12: "But everything [man and woman] comes from God."
To cap it off, whereas v.3-10 argue for women to cover their heads, Paul argues that women don't need a covering: "for long hair is given to her as a covering. If anyone wants to be contentious about this, we have no other practice - nor do the churches of God."
Conventionally, most get around the women and head covering issue by contextualising these verses and saying that it was simply the practice of the early church, no longer relevant today. However, by reading v3-10 as Paul quoting the common view which he then attacks, it makes sense of the way he then contradicts these verses. It also makes sense of the wider argument that Paul is making about freedom from the law.
Maybe when Paul says "judge for yourselves: is it proper for a woman to pray to God with her head uncovered?" we should assume his answer to the question is 'yes' and the early church was more radical than some think!