  We investigate the relationship between two properties of quantum
transformations often studied in popular subtheories of quantum theory:
covariance of the Wigner representation of the theory and the existence of a
transformation noncontextual ontological model of the theory. We consider
subtheories of quantum theory specified by a set of states, measurements and
transformations, defined specifying a group of unitaries, that map between
states (and measurements) within the subtheory. We show that if there exists a
Wigner representation of the subtheory which is covariant under the group of
unitaries defining the set of transformations then the subtheory admits of a
transformation noncontextual ontological model. We provide some concrete
arguments to conjecture that the converse statement also holds provided that
the underlying ontological model is the one given by the Wigner representation.
In addition, we investigate the relationships of covariance and transformation
noncontextuality with the existence of a quasiprobability distribution for the
theory that represents the transformations as positivity preserving maps. We
conclude that covariance implies transformation noncontextuality, which implies
positivity preservation.
