Fourier-like transformation that maps polynomials to polynomials

Question

For a nice function $f:\mathbb{R}\to \mathbb{R}$, define $T(f)$ as $$ T(f)(x) = \int_{-\infty}^{\infty} e^{\pi(x^{2}- y^{2})} \cos (2\pi x y) f(y)dy $$ One can check that, if $f(x)$ is an (even) polynomial, then $T(f)(x)$ is also an even polynomial of same degree by writing $T$ as $$ T(f)(x) = \int_{-\infty}^{\infty} e^{-\pi (y-ix)^{2}}f(y)dy = \int_{-\infty}^{\infty} e^{-\pi y^{2}}f(y+ix)dy $$ Actually, if $f(x)$ is an even function (not necessarily polynomial), then we can write $T$ as $$ T(f)(x) = e^{\pi x^{2}} \int_{-\infty}^{\infty} e^{2\pi i x y }e^{-\pi y^{2}} f(y)dy = (\mathcal{E} \circ \mathcal{F} \circ \mathcal{E}^{-1})(f)(x) $$ where $\mathcal{E}$ is a linear map $$ \mathcal{E}(f)(x) = e^{\pi x^{2}}f(x) $$ and $\mathcal{F}$ is the Fourier transform $$ \mathcal{F}(f)(x) = \int_{-\infty}^{\infty} e^{-2\pi i x y}f(y)dy $$ (Note that $(\mathcal{F} \circ \mathcal{F})(f)(x) = f(-x) = f(x)$ for even $f$, so $\mathcal{F}^{-1} = \mathcal{F}$.) Hence, $T$ is actually similar to the Fourier transform on the space of nice even functions. I'm curious if this $T$ is essentially the only transformation that similar to Fourier transform and sends even polynomial to the even polynomial of same degree. Especially, I want to know existence of other kinds of function like $e^{\pi x^{2}}$ that satisfies similar properties. When I tried $e^{\pi x^{4}}$ instead of $e^{\pi x^{2}}$, Mathematica tells me that the result was negative. (It sends polynomial to some complicated function that contains Gamma function.)