I've heard the phrase "mutual submission" used to describe the message of Ephesians 5:21: Submit to one another out of reverence for Christ.
This phrase always reminds me of a pair of Looney Tunes cartoon characters. They are chipmunks who speak with what I thought as a child was an English accent. But, their speech can more accurately be described as "high society" or genteel. Whenever the pair approached something they could not do together, like passing through a narrow opening, one would motion toward the door with his palm up and say to the other, "After you." The other would then reply, "No, after you." The first would then say, "You're too kind. Please, go ahead." This went on for a while until, finally, one would say, "Let's go together, shall we?" The other would reply, "Oh what a wonderful idea. Yes, I think we shall." Then, they would both painfully squeeze through the door together.
But even this is an understated demonstration of "mutual submission". Truly mutual submission would require us each to obey every other. This would result in near chaos as we, in our flesh, would compete to be first to give a command so that the other would be required to submit rather than vice versa. In fact, "mutual submission" is no less an oxymoron than "partially surrounded", "jumbo shrimp", or "six-month anniversary".
While I have heard it said, by those who have yet to be sanctified to the extent that they do not despise the word of God, that the subsequent verses confirm the fantasy of "mutual submission", in fact, these verses decisively demonstrate precisely the opposite.
The following verses, 5:22 - 6:9, are examples in which submission is already expected and part of the society of that time. There is nothing new in the fact of submission. The command is in how to submit. The examples give us details we can follow to give us an idea what it means to submit well. We are also encouraged and reminded in each example that our command is not merely to submit to the one who is in authority over us, but that both we in submission and those in authority are under the authority of Jesus, Lord over all.
The submission referenced in this passage is not mutual. No submission can be mutual. There is no such thing. Neither is the teaching here primarily about submission. But relationships wherein submission must occur are revealed as a context for the glory of God.