How does one swap two panes in Tmux?

The swap-pane command can do this for you. The { and } keys are bound to swap-pane -U and swap-pane -D in the default configuration.

So, to effect your desired change, you can probably use Prefix { when you are in the right pane (or Prefix } if you are in the left pane).

The -U and -D refer to “up” and “down” in the pane index order (“up” is the same direction that Prefix o moves across panes). You can see the pane indices with display-panes (Prefix q, by default).

You can hit Ctrl b and keep holding down Ctrl while hitting o. This will rotate all existing panes around, so in your case it will swap the only two existing panes.

Ctrl b + Alt o rotates the opposite way (useful when you have more than two panes).

The most precise control you can have is by using the command swap-pane directly. This is not so difficult to do:

1. ctrl-b q shows you the "ID" for each pane in current window - remember the two panes you want to swap. Let's say they're 3 and 5.
2. ctrl-b : to activate the tmux command line. Then issue command swap-pane -s 3 -t 5.

Note that you have auto-completion when typing commands. Also you can search for syntax of a command directly from tmux manpage. That's how I learn the syntax for swap-pane.