  The spherical reduction of the rational Calogero model (of type $A_{n-1}$ and
after removing the center of mass) is considered as a maximally superintegrable
quantum system, which describes a particle on the $(n{-}2)$-sphere subject to a
very particular potential. We present a detailed analysis of the simplest
non-separable case, $n{=}4$, whose potential is singular at the edges of a
spherical tetrahexahedron. A complete set of independent conserved charges and
of Hamiltonian intertwiners is constructed, and their algebra is elucidated.
They arise from the ring of polynomials in Dunkl-deformed angular momenta, by
classifying the subspaces invariant and antiinvariant under all Weyl
reflections, respectively.
