The stretching of fluid elements and the dynamics of mixing are studied for a variety of polymer solutions in nearly two-dimensional magnetically driven flows, in order to distinguish between the effects of viscoelasticity and shear thinning. Viscoelasticity alone is found to suppress stretching and mixing mildly, in agreement with some previous experiments on time-periodic flows. On the other hand, the presence of shear thinning viscosity (especially when coupled with elasticity) produces a dramatic enhancement in stretching and mixing compared to a Newtonian solution at the same Reynolds number. In order to understand this observation, we study the velocity field separately in the sheared and elongational regions of the flow for various polymer solutions. We demonstrate that the enhancement is accompanied by a breaking of time-reversal symmetry of the particle trajectories, on the average. Finally, we discuss possible causes for the time lags leading to this temporal symmetry breaking, and the resulting enhanced mixing.