Disclaimer: I am aware of the other biblical verses that assign an equal status to the man/woman or husband/wife. And I do wish that the Christian churches had focused more on something like Galatians 3:28, with everyone the same in Christ. Unfortunately, it hasn't, and the Ephesians verses is one of the reasons why.
The reason why I'm not addressing those other verses is because they seemed to get used a lot to soften what's going on in this Ephesians verse. Ephesians says what it says, and below are all the things I found buried in the verse. Softening them, or using another verse, does not eradicate what is in the Ephesians verse itself.
Yes, I'm aware of the irony, given that I don't hold God ordering much of what happened in the Tanakh, but rather people struggling to understand God. But Christianity tends not to focus on those, because Jesus set a new era. One of the verses in this new era is the Ephesians letter.
... if I was someone who believed that there were no such things as coincidences, I'd start questioning why I'm seeing Ephesians 5:22-34 pop up a lot.
It's a familiar verse, with advising women to be subject to their husbands. It's a verse used to show how women are considered secondary in the Bible. The common response to this is that the verse goes onto to tell men to love their wives as Christ loved the church and gave himself up, and for men to love their wives as themselves. It's "mutual submission."
I have a few difficulties with that response
1) There is a definite hierarchy established in this verse. Man is head of the woman, just as Christ is head of the church, so as the church submits to Christ, so must the woman submit to the man. The basic principle behind Christ being in authority is that Christ knows all, and humans are to bow to his will. In this comparison, it's establishing that the man knows better than the woman, and ultimately the woman must bow to the man's will.
2) Nowhere does it say for the man to submit to the wife as well. It says for the man to love his wife as Christ loves the church, but the man does not submit to the wife. Ever. It ends with saying that man must love his wife, but the woman must respect her husband. Now, it could be argued that "love" automatically entitles the woman to submission and respect, but that's an inference, and that depends on the definition of "love." For some men, loving their wives means they completely control their wives, because their wives aren't smart enough to guide their own lives. And this is not a "radical" view for Christianity -- from what I've seen of the early church fathers, they weren't all that thrilled with women as a whole.
3) The man is to love his wife as he loves his body. This can too easily reduce a woman to "property," much like the man's body is his own property. It is also saying that a man is to love his wife as his own self. However, self-love can often be egotistical, and self-focused. My version even says that "In loving his wife a man loves himself." In a lot of ways, the woman is almost seen as the extension of the man.
4) Only the man is compared to Christ. The woman is compared to the church. The church and Christ are not equal, so under this comparison, the man and woman are not equal, either. As the church is subject to Christ, so is the woman subject to the man. And how often do we hear that is no longer the person who lives in the body, but Christ? If that's applied here, the woman is not independent, but again -- seen as the extension of the man. The verse does go on to describe how Christ gave himself up for the church and cleansed it. But that comparison leaves us with the man needing to "save" the woman and presenting her with no stain or wrinkle. Plus, the whole purpose of Christ was that man was incapable of saving himself, so God had to step in. In the comparison, woman is incapable of overseeing herself, so the man has to step in.
I've also seen a response for this that the man and woman are equal, they just have separate roles. Yet that strays too far to the "separate but equal" line of thought, and history has shown how well that's worked.
Overall? I still see this passage as a means of controlling women. They aren't given an identity in this passage like men are.
Edited to add:
Another interesting thing I've noticed about this verse -- the wife isn't called to love the husband, simply submit and respect (Now, one can say that both are indications of love. But I also have a healthy respect for the Great White Shark, and not because of any love, but because of how quickly it can kill me. Submission and respect are not synonymous with love). If I'm remembering my history correctly, weren't there a few Greek philosophers that debated if women even had souls the way men did?
This verse almost seems to be classifying love in two different ways. Men can love, and women ... can't? Or have a substandard sort of love? It just seems to assign very clinical aspects to what the wife should do, with almost no emotion involved. It's like there's nothing of the soul involved, in what the wife is called to do. Which, in a lot of ways, would work really well with saying that the husband should love the wife as he loves his own body.