Using a small-deformation expansion and numerical simulations we study stationary shapes of viscous drops in two-dimensional linear Stokes flows with nonzero vorticity. We show that high-viscosity drops in flows with vorticity magnitude β« 1 have two branches of stable stationary states. One branch corresponds to nearly spherical drops stabilized primarily by rotation, and the other to elongated drops stabilized primarily by capillary forces. For drop-to-continuous-phase viscosity ratios beyond a critical value λc, the rotationally stabilized solution exists in the absence of capillary stresses, because the rate of drop deformation (but not rotation) decreases with drop viscosity. We show that λc = 10β-2, and the capillary stresses required for drop stability vanish at λc with exponent 1/2, as required by flow-reversal symmetry.