  We prove two results about Wigner distributions. Firstly, that the Wigner
transform is the only sesquilinear map ${\mathcal S}(\mathbb{R}^n) \times
{\mathcal S}(\mathbb{R}^n) \to {\mathcal S}(\mathbb{R}^{2n})$ which is bounded
and covariant under phase-space translations and linear symplectomorphisms.
Consequently, the Wigner distributions form the only set of quasidistributions
which is invariant under linear symplectic transformations. Secondly, we prove
that the maximal group of (linear or non-linear) coordinate transformations
that preserves the set of (pure or mixed) Wigner distributions consists of the
translations and the linear symplectic and antisymplectic transformations.
