Activity of sensory neurons is driven not only by stimuli but also by the behavior of the animal. For example, in the primary visual cortex (V1) of a mouse, the gain of sensory responses is modulated by the speed of movement [1]. On average, the response strength increases with running speed - this effect is prominent and systematic. Individual neurons can, however, reveal distinctive patterns of gain modulation. This diversity raises questions about how nuanced the impact of behavior on sensory coding is, whether it is limited to coarse gain changes, and what its computational function might be. To address these questions, we systematically characterized the joint sensory and behavioral tuning of neurons in the mouse V1. To this end, we used calcium imaging of approximately 7000 V1 neurons in behaving mice, made available by the Allen Brain Observatory [2, 3]. We focused on the relationship between the direction of visual motion and running speed - prominent sensory and behavioral variables. To extend our analysis beyond the mere correlation of response magnitude with running speed, we developed a statistical model of joint velocity-orientation tuning. The model assumes that the tuning preference of each neuron can be decomposed into direction-selective subunits that are independently modulated by velocity. By fitting the model to individual neurons, we inferred the most likely patterns of subunit modulation by locomotion state for each neuron. Our initial analysis revealed a rich diversity of previously undescribed behavioral modulations of sensory code in V1. In addition to a variety of non-monotonic gain modulations reminiscent of effects reported previously [4], we identified potentially more intricate tuning dependencies, such as shifts of preferred stimulus direction at specific velocities or changes between orientation and direction selectivity. Furthermore, preliminary information-theoretic analysis of model parameters suggests that direction and orientation selective V1 neurons allocate a substantial fraction of their resources to encode speed of movement in addition to sensory inputs. Overall, our results suggest that the primary visual cortex is not solely modulated by locomotion speed but encodes this variable jointly with stimulus features. It thus might instantiate a complex representation of the external environment and the organism's behavioral state.