  In this note we prove that a finite family $\{X_1,\dots,X_d\}$ of real r.v.'s
that is exchangeable and such that $(X_1,\dots,X_d)$ is invariant with respect
to a subgroup of $SO(d)$ acting irreducibly, is actually invariant with respect
to the action of the full group $SO(d)$. Three immediate consequences are
deduced: a characterization of isotropic spherical random eigenfunctions whose
Fourier coefficients are exchangeable, an extension of Bernstein's
characterization of the Gaussian and a characterization of the Lebesgue measure
on the sphere.
