  We show that the cross Wigner function can be written in the form $W(\psi,
\phi)= \hat S (\psi \otimes \overline{\hat\phi})$ where ${\hat\phi}$ is the
Fourier transform of $\phi$ and $\hat S$ is a metaplectic operator that
projects onto a linear symplectomorphism $S$ consisting of a rotation along an
ellipse in phase space (or in the time-frequency space). This formulation can
be extended to generic Weyl symbols and yields an interesting fractional
generalization of the Weyl-Wigner formalism. It also provides a suitable
approach to study the Bopp phase space representation of quantum mechanics,
familiar from deformation quantization. Using the "metaplectic formulation" of
the Wigner transform we construct a complete set of intertwiners relating the
Weyl and the Bopp pseudo-differential operators. This is an important result
that allows us to prove the spectral and dynamical equivalence of the
Schr\"odinger and the Bopp representations of quantum mechanics.
