  Odd viscosity is a property of chiral active fluids with broken time-reversal
and parity symmetries. We show that the flow of such a fluid around a rotating
axisymmetric body is exactly solvable and use this solution to determine the
orientational dynamics of surface-driven microswimmers. Swimmers with a
force-dipole moment exhibit precession around the axis of the odd viscosity. In
addition, pushers show bimodal chirotaxis, i.e., alignment parallel or
antiparallel to the axis, while pullers orbit in a plane perpendicular to it. A
chiral swimmer that itself has a broken parity symmetry can exhibit unimodal
chirotaxis and always align in the same direction.
