Synchronous movements of the entire face, from grimacing to chewing, offer significant insights into internal states. Mice, with discernible facial responses and evolutionarily conserved mammalian facial movement control circuits, provide an ideal model to unravel the link between facial movement and internal physiological states in mammals. While complex facial movements are orchestrated synchronous activity of muscles spread throughout the face, existing frameworks lack the spatial or temporal resolution to sensitively track motion of the entire mouse face, due to its small and conical form factor. We introduce Cheese3D, a computer vision system that first captures high-speed motion of the entire mouse face (including ears, eyes, whisker pad, mouth, while covering both sides of the face) using a calibrated six-camera array. Adapting components from existing markerless pose estimation tools, we carefully designed a hardware and software pipeline to create a unified 3D view of the mouse face, and introduced technical advancements to reduce keypoint jitters common to existing tools, hence increasing resolution and sensitivity necessary to study subtle and rapid mouse facial movements. By tracking motion in 3D, our interpretable framework extracts dynamics of anatomically-meaningful facial features in absolute world units at sub-millimeter precision. Recent improvements to Cheese3D combine these rich behavioral dynamics with temporally synchronized neural recordings from a range of sources---in vivo electrophysiology, in vivo electromyography (EMG), and electroencephalography (EEG). These features allow Cheese3D to provide clear physiological insights, as shown by proof-of-principle experiments inferring muscle anatomy from fast chewing motions, capturing subtle facial twitches under electrical stimulation of facial motor neurons, and tracking anesthesia depth using small, slow-moving changes to the mouse eyes. Cheese3D can serve as a discovery tool that renders facial movements highly interpretable as a readout of otherwise hidden internal states.