I do both. Obvious divisions can be done in the fall. Some of the more subtle ones are easier to do in the spring when the eyes become really visible, that might not have been visible in the fall.

... just my two cents worth.

I think one of the key things to dividing is having visible eyes...knowing which tubers are duds and which will produce a new flower.

Last year I lifted my tuber clumps shortly after cutting the stems, which meant I had not left them in to allow the eyes to form. When I divided my tubers I had nothing to go on, and many of them ended up not propagating (my fault!)

Everything I've read says to cut the stalks and then leave the clump in the ground for a week or two, depending on frost conditions. You may want to cover the ground with straw or some other means of keeping the tubers from freezing while this eyeing up occurs, or, heaven forbid, cut the stalks sooner...(I wouldn't!)

If you can't leave them in the ground to eye up, then I would make as few divisions as possible in the fall. Divide only to make them manageable in storage, and ensure you don't have any soil left in the clump so as to avoid rotting. Then you can make further divisions in the spring once you've brought them out and they eye up.

Cheers,
Russ