  CeRh$_2$As$_2$ is a new multiphase superconductor with strong suggestions for
an additional itinerant multipolar ordered phase. The modeling of the low
temperature properties of this heavy fermion compound requires a quartet
Ce$^{3+}$ crystal-field ground state. Here we provide the evidence for the
formation of such a quartet state using x-ray spectroscopy. Core-level
photoelectron and x-ray absorption spectroscopy confirm the presence of Kondo
hybridization in CeRh$_2$As$_2$. The temperature dependence of the linear
dichroism unambiguously reveils the impact of Kondo physics for coupling the
Kramer's doublets into an effective quasi-quartet. Non-resonant inelastic x-ray
scattering data find that the $|\Gamma_7^- \rangle$ state with its lobes along
the 110 direction of the tetragonal structure ($xy$ orientation) contributes
most to the multi-orbital ground state of CeRh$_2$As$_2$.
