I am suggesting a chair rotation for a department in sciences. I was wondering about pitfalls and advantages, things to watch out for, best practices and general advice. If you have organized such an an arrangement or have been part of one I would like to hear your recommendations.
I've been involved in a few. It's a good idea because it gives everyone a chance to participate in a leadership role. The biggest thing you can do is get organized -- ideally through documentation but maybe with something as simple as having outgoing chairs train incomings. It doesn't matter much how you rotate although at the beginning you'll probably want to start with more experienced faculty. To that same point, there might be some faculty who are not be capable of taking on the role, for whatever reason. It does no one any good to utilize these people just because it's "their turn." Good luck!
There are arrangements with a strict rotation and arrangements in which the next chair is nominated by a vote of the faculty. In either case, a higher level dean will probably have to approve the appointment. There are typically some faculty who simply shouldn't be chair for various reasons- don't lock yourself into a strict rotation.
As others have "answered" and "commented", the idea of rotation certainly has the plus of keeping people involved, and reducing the feeling of "status/hierarchy", which can become invidious.
To my mind, one key diagnostic issue is about whether your dept head is a "strong" or "weak" head. That is, if they can (if they want...) make unilateral decisions, or even simply control the agenda of faculty meetings, you'd want a sensible, balanced person in the role, obviously. On the other hand, if the chair/head mostly just signs papers and presides over faculty meetings, then, sure, anyone can do the job.
Also, I do think that there is a pitfall for mathematicians and other academics, namely, thinking that "administration" is a job that anyone can do... It is certainly true that it's a different job than "research" or "teaching" (or even refereeing papers), but in any sort of serious/large situation it is a "skill position".
And, after all, not only are not all faculty really ideal for such positions, but would often be completely happy to be left out of such a rotation, since they just want to do their research and teaching. :)
TLDR: if you have "strong" chair/head, don't have everyone rotate through, because it can make a mess. This may be an argument in favor of not having a "strong" chair/head, ... but, at my university, authority is very hierarchical... faculty only recommend upward, and all decisions about hiring/firing and money actually "descend from above". :)