  We study the micorscopic origin of the multiple superconducting and magnetic
phases observed in CeRh$_2$As$_2$. We exploit the existence of a van Hove
singularity enforced by the nonsymmorphic symmetry to conduct a renormalization
group analysis. When Fermi-surface nesting is strong, we find two
closely-competing superconducting states with opposite parities, as well as an
instability towards specific spin-density wave states, consistent with key
features of the phase diagram of CeRh$_2$As$_2$.
