Answer by Funktorality for Nontrivial invariant transformations for heat equations

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!