  The nondissipative (Hall) viscosity is known to play an interesting role in
two-dimensional (2D) topological states of matter, in the hydrodynamic regime
of correlated materials, and in classical active fluids with broken
time-reversal symmetry (TRS). However, generalizations of these effects to 3D
have remained elusive. In this work, we address this question by studying the
Hall viscoelastic response of 3D crystals. We show that for systems with
tetrahedral symmetries, there exist new, intrinsically 3D Hall viscosity
coefficients that cannot be obtained via a reduction to a quasi-2D system. To
study these coefficients, we specialize to a theoretically and experimentally
motivated tight-binding model for a chiral magnetic metal in (magnetic) space
group [(M)SG] P2$_1$3 (No. 198.9), a nonpolar group of recent experimental
interest that hosts both chiral magnets and topological semimetals (TSMs).
Using the Kubo formula for viscosity, we compute two forms of the Hall
viscosity, phonon and ``momentum'' (conventional) and show that for the
tight-binding model we consider, both forms realize the novel cubic Hall
viscosity. We conclude by discussing the implication of our results for
transport in 2D magnetic metals and discuss some candidate materials in which
these effects may be observed.
