  We consider the spherical reduction of the rational Calogero model (of type
$A_{n-1}$, without the center of mass) as a maximally superintegrable quantum
system. It describes a particle on the $(n{-}2)$-sphere in a very special
potential. A detailed analysis is provided of the simplest non-separable case,
$n{=}4$, whose potential blows up at the edges of a spherical tetrahexahedron,
tesselating the two-sphere into 24 identical right isosceles spherical
triangles in which the particle is trapped. We construct a complete set of
independent conserved charges and of Hamiltonian intertwiners and elucidate
their algebra. The key structure is the ring of polynomials in Dunkl-deformed
angular momenta, in particular the subspaces invariant and antiinvariant under
all Weyl reflections, respectively.
