[en] The authors consider the group Esup((2)) of symmetries with respect to points ('inversions') and displacements of phase space. A Wigner-Weyl system is defined as a projective representation of this group; it is a proper extension of a Weyl system. The authors derive the basic properties of Wigner-Weyl systems and show: (i) Their use clarifies the role of symplectic Fourier transform in the Weyl correspondence. (ii) The quasiprobability density of Wigner can be written in an intrinsic, symplectically covariant way as a matrix element of Wigner operators. (Auth.)