The network of grid cells in the medial entorhinal cortex (MEC) forms a fixed reference frame for mapping physical space. The mechanistic origin of the grid representation is unknown, but continuous attractor network models explain multiple fundamental features of grid cell activity. An untested prediction of these models is that the grid cell network should exhibit an activity correlation structure that transcends behavioral states. By recording from MEC cell ensembles during navigation and sleep, we found that spatial phase offsets of grid cells predict arousal-state-independent spike rate correlations. Similarly, state-invariant correlations between conjunctive grid-head direction and pure head direction cells were predicted by their head direction tuning offsets during awake behavior. Grid cells were only weakly correlated across grid modules, and module scale relationships disintegrated during slow-wave sleep, suggesting that grid modules function as independent attractor networks. Collectively, our observations imply that network states in MEC are expressed universally across brain and behavior states.