It means you play through the four characters in order in four successive runs, e.g. ironclad -> silent -> defect -> watcher, rather than e.g. just playing the watcher (the character generally considered the easiest to win with at high skill levels) 4 times in a row.

I think it's changing characters on each run, 'rotating' through the lot.

That was my best guess too. But I feel like there could easily be some other meaning that makes sense to someone who has obsessively played thousands of games of Spire.

I googled it a bit and that seems to be it and what made sense to me, having obsessively played probably only hundreds of games of Spire.