> If you have -> → and <- ← you’re fine, but when you add <-> ↔, I can’t find any way of using the <- sequence any more.

X11 is likely walking a tree of .XCompose entries with each keypress. Once it gets to '<' and '-' it finds '←' and does not continue to consider your next '>'. So, you need to provide a way to walk a different path.

This works for me.

    <Multi_key> <less> <period> <greater> : "↔"

In general we must consider the entirety of .XCompose when choosing new compose key bindings. Maybe there is some utility to help with that. For me, I removed 98% of the default Compose file entries which makes manual checking feasible.

There is no X11 involved here, and even on systems running an X server instead of Wayland, judging by the symptoms I’ve seen, the X server isn’t actually involved in interpreting Compose sequences—each app implements the whole lot itself, and judging by the inconsistencies, not all are using the same library for it.

Some only let Compose < - (←) work, stopping and preventing Compose < - > (↔) from working. Others, if I remember correctly, let Compose < - Enter work to get ←.

Once an Input Method is involved, it can handle the Compose key, and that’s what fcitx5 is doing for me now, so that everything’s behaving the same… but that “same” is not what I reckon it should be.