Hilbert's third problem asks for a classification of polytopes up to scissors congruence. Complementary to this one could ask about the group of scissors automorphisms of a fixed polytope; ways to cut a polytope and reassemble it from the pieces. In joint work with Lemann, Malkiewich, Miller, and Sroka, we compute the homology of these groups in terms of certain Thom spectra. Our techniques also apply to variants of these groups that have been studied in geometry group theory and dynamical systems, such as interval exchange groups and Thompson groups. Indeed, our work generalises that of Szymik and Wahl on the latter.