  We theoretically investigate the $d^4$ (Cr$^{2+}$) compound BaCr$_2$As$_2$
and show that, despite non-negligible differences in the electronic structure,
its many-body physics mirrors that of BaFe$_2$As$_2$, which has instead a $d^6$
(Fe$^{2+}$) configuration. This reflects a symmetry of the electron correlation
effects around the half-filled $d^5$ Mott insulating state. The experimentally
known metallic antiferromagnetic phase is correctly modeled by dynamical
mean-field theory, and for realistic values of the interaction it shows a
moderate mass enhancement of order $\sim$2. This value decreases if the ordered
moment grows as a result of a stronger interaction. The antiferromagnetic phase
diagram for this $d^4$ shows similarities with that calculated for the $d^6$
systems. Correspondingly, in the paramagnetic phase the influence of the
half-filled Mott insulator shows up as a crossover from a weakly correlated to
an orbitally differentiated "Hund's metal" phase which reflects an analogous
phenomenon in $d^6$ iron compounds including a strong enhancement of the
compressibility in a zone just inside the frontier between the normal and the
Hund's metal. The experimental evidence and our theoretical description place
BaCr$_2$As$_2$ at interaction strength slightly below the crossover which
implies that negative pressures and/or electron doping (e.g. Cr $\rightarrow$
Mn,Fe or Ba $\rightarrow$ Sc,Y,La) might strongly enhance the compressibility,
thereby possibly inducing a pairing instability in this non-superconducting
compound.
