We investigated the inelastic hard-disk gas sheared by two parallel bumpy walls (Couette flow). In our molecular dynamic simulations we found a sensitivity to the asymmetries of the initial particle positions and velocities and an asymmetric stationary state, where the deviation from (anti)symmetric hydrodynamic fields is stronger as the normal restitution coefficient decreases. For better understanding of this sensitivity we carried out a linear stability analysis of the former kinetic theoretical solution [J. T. Jenkins and M. W. Richman, J. Fluid. Mech. 171, 53 (1986)] and found it to be unstable. The effect of this asymmetry on the self-diffusion coefficient is also discussed.