Yesterday I did a satirical post on the Christian headcovering movement, but now I'll do a serious post on 1 Cor 11.
i) I'd note in passing that the best current commentaries on 1 Corinthians are by Fitzmyer, Ciampa/Rosner, and Thiselton. There are also some fine forthcoming commentaries in the pipeline. Gordon Fee is revising his classic commentary. The late E. E. Ellis was writing a commentary. I don't know how close to completion he was. Bruce Winter has a commentary that's due out soon.
ii)1 Cor 11:2-16 is a tricky text to navigate. Paul alternates between literal and figurative senses of "head." There are subtextual allusions to Gen 1-2, Exod 34:33-34, and Ps 8:5. There's an enigmatic reference to angels. There's the question of whether Paul is referring to women generally, or married women in particular.
Although this is ancillary to the main point of my post, I think Paul's cryptic reference to angels (11:10) probably picks up on Ps 8:5. He already alluded to that passage in v7. So I think v10 carries that over and rounds out that theme. And that's seconded by the angelophany in Exod 34.
iii) A particular complication is the discussion of dress codes. In the nature of the case, fashion trends rapidly go in and out of fashion. Our surviving background information isn't sufficiently fine-tuned to give us a precise bead on dress codes in Corinth at the time of writing (c. 53-54). So the modern reader is somewhat in the dark.
As a result, there's a danger of overworking what background information we have. Scholars can only use what background information is available to them, so there's a temptation of forcing a fit between the text and the available background information, which may be adventitious.
iv) Due to our relative ignorance of Corinthian fashion at the time of writing, I don't think it's possible to fully reconstruct the setting. Complete understanding will elude us. Consider this in reverse. Imagine if a magazine article with references to cybergoth fashion went through the time machine and landed in the hands of a 1C Corinthian reader.
The Bible is written for the benefit of more than one audience. There's the immediate audience, and then there's posterity. At this distance, there's some loss of specific understanding.
v) Fortunately, I don't think that's debilitating. Although our understanding of Corinthian fashion is deficient, we can still get the gist of what Paul is saying, for we have an analogous understanding of fashion. Hairstyles and styles of dress send a multiplicity of messages. They can signal sexual availability or unavailability, age, gender, region, ethnicity, social class, subculture, employment, &c.
A man may dress like a cowboy because he is a cowboy, or because he lives in a part of the country where that attire is customary, or because that projects a manly image. A man or woman may dress in motorbike regalia (a la Hell's Angels) as a countercultural statement. That's simultaneously expresses solidarity with a subculture and nonconformity in relation to the mainstream culture. Conversely, some people practice a studied indifference to fashion trends. That, itself, is sending a message.
vi) Some readers find Paul's discussion silly or arbitrary. But that's because fashion itself can be silly or arbitrary. Paul didn't create that state of affairs. The challenge is how a Christian minority of dubious legal standing should function in an overwhelmingly pagan city. Judging people by how they dress is often unfair, but because that's inevitable, it's something we have to take into account. Like it or not, that's the reality.
vii) Notice that Paul isn't talking about every woman who attends a worship service. Rather, he singles out a subset of women who actively participate ("pray and prophesy") in a worship service (11:5).
Now, to my knowledge, Christians who think women should wear headcoverings in church are usually cessationists and complementarians. If, however, you're a cessationist, then no woman would ever have occasion to prophesy in church, since you believe the gift of prophecy lapsed after the NT era. To that extent, the question of headcoverings is mooted by your cessationism.
You can redefine "prophecy" in some idiosyncratic, non-charismatic sense, but that isn't what Paul means.
Likewise, prayer has reference to praying aloud in public worship. But some complementarians don't think women should speak in church, in mixed company. It's not their place to address the congregation, or lead in worship. If that's your position, then the question of headcoverings is mooted by your complementarianism.
For Paul's requirement doesn't apply to women who aren't praying or prophesying. It doesn't apply to women who sit quietly in the pews, listening to elders preach, pray, or recite the Bible. It's only dealing with active, not passive, worshipers.
So it seems to be that the headcovering movement is ironically at odds with its prooftext. But there may be exceptions.
viii) It's important to distinguish the culturalbound elements of Paul's discussion from the transcultural elements. Male headship is a transcultural principle, grounded in the "creation order."
By contrast, headcoverings symbolize the principle. Headcoverings are not, in themselves, a timeless principle. Rather, their function is emblematic. They merely illustrate the principle. That distinction ought to be clear from Paul's own discussion.
ix) And symbolism is often variable. Fashion that's "shameful" in one time or place may not be "shameful" in another, or vice versa. Head coverings don't have the same social or symbolic connotations in 21C America that they had in 1C Corinth.
BTW, Paul's honor/shame dialectic probably trades on the honor/glory motif in Ps 8:5.
Now, some Christians may favor head coverings as a countercultural statement. But whatever the merits of a countercultural statement in its own right, don't confuse that with obeying 1 Cor 11. We don't obey Scripture by applying commands out of context, but by applying commands to comparable situations. Reproducing a 1C dress code that doesn't have the same significance in the 21C isn't honoring 1 Cor 11. To take a comparison, you don't obey Deut 22:8 by putting a railing around a gabled roof.