  We study the covariance property of quadratic time-frequency distributions
with respect to the action of the extended symplectic group. We show how
covariance is related, and in fact in competition, with the possibility of
damping the interferences which arise due to the quadratic nature of the
distributions. We also show that the well known fully covariance property of
the Wigner distribution in fact characterizes it (up to a constant factor)
among the quadratic distributions $L^{2}(\mathbb{R}^{n})\rightarrow C_{0}({
\mathbb{R}^{2n}})$. A similar characterization for the closely related Weyl
transform is given as well. The results are illustrated by several numerical
experiments for the Wigner and Born-Jordan distributions of the sum of four
Gaussian functions in the so-called "diamond configuration".
