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

Quantum mappings acting by coordinate transformations on Wigner distributions