We consider spin-vorticity coupling-the generation of spin polarization by vorticity-in viscous two-dimensional electron systems with spin-orbit coupling. We first derive hydrodynamic equations for spin and momentum densities in which their mutual coupling is determined by the rotational viscosity. We then calculate the rotational viscosity microscopically in the limits of weak and strong spin-orbit coupling. We provide estimates that show that the spin-orbit coupling achieved in recent experiments is strong enough for the spin-vorticity coupling to be observed. On the one hand, this coupling provides a way to image viscous electron flows by imaging spin densities. On the other hand, we show that the spin polarization generated by spin-vorticity coupling in the hydrodynamic regime can, in principle, be much larger than that generated, e.g. by the spin Hall effect, in the diffusive regime.