  The richness of quantum theory's reversible dynamics is one of its unique
operational characteristics, with recent results suggesting deep links between
the theory's reversible dynamics, its local state space and the degree of
non-locality it permits. We explore the delicate interplay between these
features, demonstrating that reversibility places strong constraints on both
the local and global state space. Firstly, we show that reversible dynamics are
trivial (composed of local transformations and permutations of subsytems) in
maximally non-local theories whose local state spaces satisfy a dichotomy
criterion; this applies to a range of operational models that have previously
been studied, such as d-dimensional "hyperballs" and almost all regular
polytope systems. By separately deriving a similar result for odd-sided
polygons, we show that classical systems are the only regular polytope state
spaces whose maximally non-local composites allow for non-trivial reversible
dynamics. Secondly, we show that non-trivial reversible dynamics do exist in
maximally non-local theories whose state spaces are reducible into two or more
smaller spaces. We conjecture that this is a necessary condition for the
existence of such dynamics, but that reversible entanglement generation remains
impossible even in this scenario.
