  Odd viscosity can emerge in 3D hydrodynamics when the time reversal symmetry
is broken and anisotropy is introduced. Its ramifications on the stability of
the prototypical Taylor-Couette flow in curved geometries have remained
unexplored. Here, we investigate the effects of odd viscosity on the stability
of Taylor-Couette flow under axisymmetric perturbations both analytically and
numerically, deriving analytically the critical Taylor number for different odd
viscosities in the narrow gap case as well as fully numerically implementing
the wide gap case. We find that the odd viscosity modifies the vortex pattern
by creating secondary vortices, and exerts an intriguing ``lever'' effect in
the stability diagram, completely suppressing the instability of Taylor-Couette
flow under axisymmetric perturbations when the odd viscosity is large,
irrespective of its sign. Our findings highlight the role of odd viscosity for
the rich flow patterns of the Taylor-Couette geometry and provide guidance for
viscometer experiments when odd viscosity is present.
