Recent work has shown that spontaneous movements are represented throughout the brain, even in sensory areas such as primary visual cortex (V1). Interference between visual stimuli and movement is minimized in V1 because they are encoded in orthogonal subspaces, but it is not known how these subspaces are formed. Here, we report that Neuropixels recordings in mouse V1 during slow-wave sleep (SWS) reveal internally generated low-dimensional manifold structure that is mostly preserved in the awake state. However, the awake state has more variance in “off-manifold” dimensions, defined as those exhibiting less variance during SWS than expected by chance. Spontaneous movements are encoded in “on-manifold” dimensions, which during SWS account for more variance than chance and exhibit multi-region coordination strikingly similar to during movement. Surprisingly, we found that natural visual scenes are encoded in both on-manifold dimensions and off-manifold dimensions. Off-manifold dimensions are underrepresented in spontaneous activity because they comprise sparse activation of neurons that are statistically unlikely to fire sparsely due to their strong coupling to the overall population. Internally-generated low-dimensional structure in V1 thus creates both a dense on-manifold subspace encoding movements and a sparse off-manifold subspace encoding stimulus features. These results reveal an unexpected link between dimensionality and sparse coding and also shed light on a simple yet overlooked mechanism by which brainwide representations of movement can coexist peacefully with local representations of task-related variables.