The character of the ion dynamics in crystalline cryolite, Na3AlF6, a model double perovskite-structured mineral, has been examined in computer simulations using a polarizable ionic potential obtained by forcefitting to ab initio electronic structure calculations. NMR studies, and conductivity measurements, have indicated a high degree of mobility, in both Na+ ion diffusion and reorientation of the AlF6 octahedral units. The simulations reproduce the low-temperature (tilted) crystal structure and the existence of a transition to a cubic structure at elevated temperatures, in agreement with diffraction measurements, though the calculated transition temperature is too low. The reorientational dynamics of the AlF6 octahedra is shown to consist of a hopping motion between the various tilted positions of the low-temperature form, even above the transition temperature. The rate of reorientation estimated by extrapolation to the temperature regime of the NMR measurements is consistent with the experimental data. In addition, we report a novel cooperative "tilt-swapping" motion of the differently tilted sublattices, just below the transition temperature. The perfect crystals show no Na+ diffusion, in apparent disagreement with observation. We argue, following previous analyses of the cryolite phase diagram, that the diffusion observed in the experimental studies is a consequence of defects that are intrinsic to the thermodynamically stable form of cryolite. By introducing defects into the simulation cell, we obtain diffusion rates that are consistent with the NMR and conductivity measurements. Finally, we demonstrate a link between diffusion of the Na+ ions and the reorientation of AlF6 units, though the correlation between the two is not very strong.