If they're like what I am, they seem to just coordinate constant staggered resets for sub-systems of language processing pipeline while keeping internal representations of inputs in half-text state so that input come back out through the pipeline in the other configurations.

That's how I anecdotally feel and interpret how my own brain appear to work, so it could be different from how interpreters work or how actual human brains work, but as far as I see it, professional simultaneous interpreters don't seem to be agnostic for relevant pairs of languages at all.