In cat visual cortex, the response of a neural population to the linear combination of two sinusoidal gratings (a plaid) can be well approximated by a weighted sum of the population responses to the individual gratings—a property we refer to as subspace invariance. We tested subspace invariance in mouse primary visual cortex by measuring the angle between the population response to a plaid and the plane spanned by the population responses to its individual components. We found robust violations of subspace invariance arising from a strong, negative correlation between the responses of neurons to individual gratings and their responses to the plaid. Contrast invariance, a special case of subspace invariance, also failed. The responses of some neurons decreased with increasing contrast, while others increased. Altogether the data show that subspace and contrast invariance do not hold in mouse primary visual cortex. These findings rule out some models of population coding, including vector averaging, some versions of normalization and temporal multiplexing.