I'm not sure how far, technically, this idea could go. But in it's simplest form it doesn't seem that much of a stretch. I'd like to see a face rotation manipulate the base sketch of the extrusion. Ideally it would also apply an angular dimension to the sketch, instead of just removing any neccessary constraints.

I don't see why in this case the sketch couldn't be modified appropriately.
In fact it should occur with a simple push/pull on a face as well.