Found! Spontaneous baby Clematis!
While pruning the Clematis, I found a small one growing about a foot in front of the parent plants (Minuet and Polish Spirit) that must have grown from a layered stem lying on the ground last year. It's been there since at least last summer, as the main part has a big bunch of basal leaves, with 1 oldish woody stem and a bunch more leaves on the end of that.
I'd like to transplant it to another place in the yard. I'm assuming I need to dig up the whole root ball and handle it like a family heirloom passed down from the Mayflower. :-) Can I just pop it in a new whole, or should I put it in a large nursery pot to hold until Fall? I want to give it the best possible start, and I don't want to leave it where it is.