We consider fluids where the particles possess a microstructure (micropolar flows) and are allowed to rotate in a two-dimensional setting. We find the asymptotic profile of the solution as the time goes to infinity. We deduce the remarkable fact that the large time behavior only depends on the kinematic viscosity, and not on the other parameters of the model (vortex-viscosity, spin viscosity and gyroviscosity). Our primary tool is a new enstrophy-like identity of independent interest, involving the difference between the fluid vorticity and the micro-angular velocity. This is joint work with L. Brandolese, V. Busuioc and C. F. Perusato.