From the structure and sequence of the piyut "Melech Elyon", it is clear that each phrase is meant to begin with the words 'Melech Elyon' and end with 'laadei ad Yimloch', and indeed it is printed that way. However, the prevailing practice is to say 'Melech Elyon' at the end of each phrase. What could be the reason for this divergence from the written form of the piyut?
Kovetz Yagdil Torah Volume 17 - page 39 says that originally there were alternate stanzas with Melech Elyon then Melech Evyon throughout the entire Piyut. He says that since the Melech Evyon verses were skipped, the congregation used to say immediately after the end of the verse Melech Elyon in order to indicate that.
Most Piyutim were designed Chazan - Kahal (that the Chazan says a part, and the Kahal's says the following part or a chorus), both for the beauty of it (as is done by Anum Zmirot, and how the Gemara and Rambam describe how Hallel and Shira is done), and for practical reasons, as most users in the olden days did not have a Machzor. Therefore the Chazan would often lead into the Kahal's part (often, starting a pasuk, which the Kahal will understand to finish, as is done in some Yom Kippur Katon Piyutim), or the Kahal will only repeat a chorus. The chorus will sometimes be before the Chazan, like Melach Elyon, and sometimes after the Chazan, like Bedil Vayovor.
What happened with time, is that the Kahal got Machzorim, and they would like to sneak in the Chazan's part. So for example by Melech Elyon, the Kahal would whisper with the Chazan his part, and then yell loudly their part, which is Melech Elyon, so eventually the two got attached. The same by Vchol Maminim, where Vchol Maminim is obviously the Kahal's response to the Chazan's previous praise, but due to each one trying to whisper the other's part, with the Kahal saying their part, and quickly saying the Chazan's next part, till it got reversed. Same is by all the rest of the Piyutim.
You can actually see this in action on Yom Kippur Katon, by the Vedil Vayovor Piyut, which most congregations still have it Chazan-Kahal, with the Kahal responding Vedil Vayovor. But you always see some people try to say along with the Chazan, and they say it quicker, so as to finish before Chazan. So they end up saying "Vedil Vayovor Rachmono Itkar lon ...". Sometimes due to noise level, the Chazan has to wait for them to finish. So there we have it in the near future, being said in reverse like most Piyutim.
Beurei Hatefillah ascribes the problem to a general printing error, much of it begun by Birnbaum's Siddur and in some sense, "enhanced" by Art Scroll.
He states that the general format of piyutim was pretty much as @lionscribe described. Most Piyutim follow an Aleph-Bet format, and the chazzan recited the Aleph line, the cong. said the Bet and Gimel, and the chazzan responded.
Another format had the chazzan read half line and the cong. responds with a whole line. This is the original format used and written for Vechol Ma'aminim. You'll see this format on the site as printed in an Amsterdam Siddur dated 1768. It is quite clearly marked what the chazzan says, and what the cong. says.
Another 20th century Siddur from Hebrew Publishing Co. 1922 follows the same layout, but is less clear. Birnbaum maintained the same layout as the Heb. Pub. siddur, but, left one instruction at the beginning that just says "Responsively". Thus, the author claims the confusion began. What does "responsively" mean, and who starts what?
Art Scroll added more to this confusion in particular with Vechol Ma'aminim by keeping Birnbaum's "responsively" instruction, and adding "The chazzan says the first verse in the same breath as part of the previous verse. Granted, this additional instruction seems unique to Vechol Ma'aminim. But, the "responsively" instruction is for all the piyutim, and that's likely where the confusion remains.
Many other siddurim I have seen place the first part of the piyut in the center of the margins, separate the other verses in 2 columns, and the last half verse is again in the center. This really confuses things.
IMO, there is no question that a version that is clearly marked with the wording chazzan and kahal (or English equivalent) would easily clarify things. I'm not sure which "modern" Siddurim are doing this and are doing correctly restoring things to their original pattern.