Collective migration of cancer cells is often interpreted using concepts derived from the physics of active matter, but the experimental evidence is mostly restricted to observations made in vitro. Here, we study collective invasion of metastatic cancer cells injected into the mouse deep dermis using intravital multiphoton microscopy combined with a skin window technique and three-dimensional quantitative image analysis. We observe a multicellular but low-cohesive migration mode characterized by rotational patterns which self-organize into antiparallel persistent tracks with orientational nematic order. We analyze the deformations induced by the cells in the extracellular matrix and find broadly distributed strain bands with a prevalence of compression. A model of active nematic hydrodynamics is able to describe several statistical features of the experimentally observed flow, suggesting that collective cancer cell invasion can be interpreted as a nematic active fluid in the turbulent regime. Our results help elucidate the migration patterns of cancer cells in vivo and provide quantitative guidance for the development of realistic in vitro and in silico models for collective cell migration.