  Quantum magnets with spin $J=2$, which arise in spin-orbit coupled Mott
insulators, can potentially display multipolar orders. We carry out an exact
diagonalization study of a simple octahedral crystal field Hamiltonian for two
electrons, incorporating spin-orbit coupling (SOC) and interactions, finding
that either explicitly including the $e_g$ orbitals, or going beyond the
rotationally invariant Coulomb interaction within the $t_{2g}$ sector, causes a
degeneracy breaking of the $J\!=\!2$ level degeneracy. This can lead to a
low-lying non-Kramers doublet carrying quadrupolar and octupolar moments and an
excited triplet which supports magnetic dipole moments, bolstering our previous
phenomenological proposal for the stabilization of ferro-octupolar order in
heavy transition metal oxides. We show that the spontaneous time-reversal
symmetry breaking due to ferro-octupolar ordering within the non-Kramers
doublet leads to electronic orbital loop currents. The resulting internal
magnetic fields can potentially explain the small fields inferred from
muon-spin relaxation ($\mu$SR) experiments on cubic $5d^2$ osmate double
perovskites Ba$_2$ZnOsO$_6$, Ba$_2$CaOsO$_6$, and Ba$_2$MgOsO$_6$, which were
previously attributed to weak dipolar magnetism. We make further predictions
for oxygen NMR experiments on these materials. We also study the reversed level
scheme, where the $J\!=\!2$ multiplet splits into a low-lying magnetic triplet
and excited non-Kramers doublet, presenting single-ion results for the magnetic
susceptibility in this case, and pointing out its possible relevance for the
rhenate Ba$_2$YReO$_6$. Our work highlights the intimate connection between the
physics of heavy transition metal oxides and that of $f$-electron based heavy
fermion compounds.
