  This note contains a new characterization of modulation spaces
$M^p(\mathbb{R}^n)$, $1\leq p\leq \infty$, by symplectic rotations. Precisely,
instead to measure the time-frequency content of a function by using
translations and modulations of a fixed window as building blocks, we use
translations and metaplectic operators corresponding to symplectic rotations.
Technically, this amounts to replace, in the computation of the
$M^p(\mathbb{R}^n)$-norm, the integral in the time-frequency plane with an
integral on $\mathbb{R}^n\times U(2n,\mathbb{R})$ with respect to a suitable
measure, $U(2n,\mathbb{R})$ being the group of symplectic rotations. More
conceptually, we are considering a sort of polar coordinates in the
time-frequency plane. In this new framework, the Gaussian invariance under
symplectic rotations yields to choose Gaussians as suitable window functions.
We also provide a similar characterization with the group $U(2n,\mathbb{R})$
being reduced to the $n$-dimensional torus $\mathbb{T}^n$.
