= Spherical inversion preserves angular derivatives
{title2=$\mathbf r\times\nabla_{\mathbf r}=\overline{\mathbf r}\times\nabla_{\overline{\mathbf r}}$}
For $\overline{\mathbf r}=a^2\mathbf r/r^2$, the <chain rule> gives $\nabla_{\mathbf r}=(a^2/r^2)(I-2\widehat{\mathbf r}\widehat{\mathbf r}^{\,T})\nabla_{\overline{\mathbf r}}$. The radial term is killed by crossing with $\mathbf r$, giving the identity. Purely angular <vector spherical harmonics> are therefore unchanged as angular basis functions. This is distinct from pushing forward a physical vector field by the inversion Jacobian.
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