Molecular dynamics simulations of pure liquid water under ambient conditions using four common empirical water models have been analyzed to determine how well the oxygen-oxygen radial distribution function, g(OO)(r), used as the sole criterion of congruence with experiment, captures variations in the actual anisotropic collective structuring for these models. The largest systematic deviations from tetrahedrality were found to be due to deformations of the angle between the two closet hydrogen bond donor neighbors, but for intrinsic geometric reasons, these were found to contribute less to g(OO)(r) than deformations of the angles between one hydrogen bond donor neighbor and one hydrogen bond acceptor neighbor. Relying exclusively on a qualitative characterization of the second peak in g(OO)(r) seems to overemphasize the differences between the structuring in some of these models.