  We consider the orientational instabilities, both homogeneous and spatially
periodic, developing in a nematic liquid crystal under rectilinear oscillatory
Couette flow for director alignment perpendicular to the flow plane. Using
numerical and analytical approaches we determine the critical amplitude of
oscillatory flow instabilities, the critical wave number and the symmetry of
the destabilizing mode. It was found, that by varying of the oscillatory flow
frequency the instability changes its temporal symmetry. The mechanism of this
transition is coupled with the inertia of the nematic fluid. We also shown that
an electric field applied to the nematic layer can induce switching between
instabilities with different spatial and temporal symmetries. The complete
phase diagram of the flow instabilities is presented
