  We develop the theory of Wigner representations for a large class of
operational theories that include both classical and quantum theory. The Wigner
representations that we introduce are a natural way to describe the theory in
terms of some fixed observables; these observables are often picked to be
position and momentum or spin observables. This allows us to introduce
symmetries which transform the outcomes of the observables used to construct
the Wigner representation; we obtain several results for when these symmetries
are well defined or when they uniquely specify the Wigner representation.
