I notice in all this discussion on adventitious buds/roots that no-one has mentioned the fact that some peonies run - sometimes quite some distance - so if the main plant is removed perhaps it is some of these runners that are noticed. A vigorous runner in my garden is tenuifolia Rubra, followed by Ellen Cowley and Nosegay. As I only move peonies when their area becomes too shady, I always try and leave the roots I break where they were so I remember better when any surprise shoots appear! Certainly many of the corals have become cut-and-come-again plants here