It is relatively easy to show that the Laplacian
$$ \Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} $$
Is the unique second order linear differential operator that is invariant under rotations in the sense that
$$ \Delta (f(R\mathbf{x})) = (\Delta f)(R\mathbf{x}). $$
The way I remember proving this was to write down a general
$$ D = \sum_i a_i \frac{\partial}{\partial x_i} + \sum_{ij} b_{ij} \frac{\partial^2}{\partial x_i \partial x_j}, $$
and then demand the invariance property. A multiple of the Laplacian will then fall out.
I am wondering if there is a way to general all such operators (up to a certain degree). Are they all powers of the Laplacian? What happens for the vector Laplacian?