Why doesn't the interviewee ever ask the interviewer a whiteboard or puzzle question, "to see how you think?"
Ask their own whiteboard question? That would be strange, as I'm a manager and don't write much code these days. But, if I ask a whiteboard question (or any tech question), asking me related questions as a means to further the discussion would be a good signal.
I'm a manager that rarely codes as well, but was still asked coding questions in an interview. It's a bit annoying, but having had managers that have never written a line of code ever, I understand wanting to verify that the manager could theoretically do the work themselves.