  The Stokes equation describes the motion of fluids when inertial forces are
negligible compared to viscous forces. In this article, we explore the
consequence of parity-violating and non-dissipative (i.e. odd) viscosities on
Stokes flows in three dimensions. Parity-violating viscosities are coefficients
of the viscosity tensor that are not invariant under mirror reflections of
space, while odd viscosities are those which do not contribute to dissipation
of mechanical energy. These viscosities can occur in systems ranging from
synthetic and biological active fluids to magnetised and rotating fluids. We
first systematically enumerate all possible parity-violating viscosities
compatible with cylindrical symmetry, highlighting their connection to
potential microscopic realizations. Then, using a combination of analytical and
numerical methods, we analyze the effects of parity-violating viscosities on
the Stokeslet solution, on the flow past a sphere or a bubble, and on
many-particle sedimentation. In all the cases we analyze, parity-violating
viscosities give rise to an azimuthal flow even when the driving force is
parallel to the axis of cylindrical symmetry. For a few sedimenting particles,
the azimuthal flow bends the trajectories compared to a traditional Stokes
flow. For a cloud of particles, the azimuthal flow impedes the transformation
of the spherical cloud into a torus and the subsequent breakup into smaller
parts that would otherwise occur. The presence of azimuthal flows in
cylindrically symmetric systems (sphere, bubble, cloud of particles) can serve
as a probe for parity-violating viscosities in experimental systems.
