Let $R_{\theta}$ be the rotation by an angle $\theta$.
Is it then true that for multi-indices $\alpha$ of fixed order $j$ and any smooth function $f$ we have
$$\sum_{\vert \alpha \vert=j}(R_{\theta}z)^{\alpha} \partial^{\alpha}f(x) = \sum_{\vert \alpha \vert=j}(z)^{\alpha} (R_{-\theta}\partial)^{\alpha}f(x)$$
It is true for $j=1$, which just follows since $R_{-\theta}$ is the adjoint map of $R_{\theta}.$