Experimental NMR measurements of the deuterium and O-17 T-1 relaxation times in deuterium-enriched liquid water have been performed. from 275 to 350 K. These relaxation times can yield rotational correlation times of appropriate molecule-fixed unit vectors if the quadrupole coupling constants and asymmetry parameters are known. We determine the latter from ab initio studies of water clusters and experimental chemical shift measurements. We find that the rotational correlation time for the OD bond vector in (D2O)-O-16 varies from 5.8 ps at 275 K to 0.86 ps at 350 K, and that the rotational correlation time for the out-of-plane vector of dilute (D2O)-O-17 in (D2O)-O-16 varies from 4.4 ps at 275 K to 0.64 ps at 350 K. These results indicate that the rotational motion of water is anisotropic. Molecular dynamics simulations of liquid water are in good agreement with these experiments at the higher temperatures, but the simulation results are considerably faster than experiment at the lower temperatures.