Calogero models are prototypes of superintegrable quantum multi-particle models. For the rational A-type case we review their infinite algebra of symmetric observables, which includes all conserved charges. For a fixed number of particles, functional dependency leads to a finite but nonlinear W algebra, for which we construct a first Casimir operator. Another well-known feature is the existence of (horizontal) intertwiners, which relate the energy spectra for integral shifts of the Calogero coupling and give rise to so-called "analytic integrability" for integer coupling values. Here, we report on our recent discovery of "vertical" intertwiners, which increase by one the number of interacting Calogero particles for any fixed integer coupling.