  Vacancy-ordered halide double perovskites hosting 4d/5d transition metals
have emerged as a distinct platform for investigating unconventional magnetism
arising out of the interplay of strong atomic spin-orbit coupling (SOC) and
Coulomb interactions. Focusing on the $d^2$ system Cs$_2$WCl$_6$, our ab initio
electronic structure calculation reveals very narrow electronic bands,
fulfilling the necessary condition to realize exotic orders. Using this input,
we solve the many-body spin-orbit coupled single-site problem by exact
diagonalization and show that the multiplet structure of Cs$_2$WCl$_6$ hosts
ground non-Kramers doublets on W, with vanishing dipole moment and a small gap
to an excited magnetic triplet. Our work provides the rationale for the
observed strong deviation from the classic Kotani behaviour in Cs$_2$WCl$_6$
for the measured temperature dependence of the magnetic moment. The non-Kramers
doublets on W exhibit non-zero quadrupolar and octupolar moments, and our
calculated two-site exchange supports the dominance of inter-site octupolar
exchange over quadrupolar interactions. We predict ferro-octupolar order with a
transition temperature $T_c \sim 5$K which may get somewhat suppressed by
quantum fluctuations and disorder; this could be tested in future
low-temperature experiments.
