= Rotation equivariance of the Fourier transform
For an orthogonal matrix $R$, a unit-Jacobian change of variables gives $\widehat{\varphi\circ R^t}=\widehat\varphi\circ R^t$. The dual definition of the <Fourier transform of a tempered distribution> consequently gives
$$
\widehat{T\circ R}=\widehat T\circ R.
$$
Since the <Fourier transform> is invertible on the <Schwartz space> and its dual, a <tempered distribution> is invariant under a rotation group exactly when its transform is invariant. This includes <radial tempered distributions>.
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