Yes, how about the Appell transform, see here.

$$v(t,x)\mapsto\Gamma(t,x)v(\frac xt,-\frac1t)$$ where $\Gamma$ is the heat kernel.

Truly nontrivial if you ask me. Of course, this inverts time and not space!

Yes, how about the Appell transform, see here.

$$v(t,x)\mapsto\Gamma(t,x)v(-\frac1t,\frac xt)$$ where $\Gamma$ is the heat kernel.

Truly nontrivial if you ask me. Of course, this inverts time and not space!