As cosmology enters its precision era, it is striking how little we know about the primordial Universe. Constraints on inflationary physics have barely advanced beyond the Planck mission, leaving key aspects of the inflationary sector, e.g. its particle content, interactions, and symmetries largely unknown. In this talk, I will discuss how the cosmic momentum field of large-scale structure encodes crucial information about the physics of inflation through its parity structure, yet its statistics remain relatively unexplored. Specifically, I will highlight its complementarity to current probes of parity violation, in particular its capability of encapsulating higher-point statistics in lower-point observables. Its ubiquity also enables powerful synergies between upcoming CMB and galaxy surveys, and I will demonstrate this with a concrete application using kinetic Sunyaev-Zel'dovich tomography.