Yes, how about the Appell transform, see [here][1].


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

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

  [1]: https://www.jstor.org/stable/1993650