  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.

Mesoscale and Nanoscale Physics