Several types of active agents self-propel by spinning around their propulsion axis, thus behaving as active screws. Examples include cytoskeletal filaments in gliding assays, magnetically driven colloidal helices, and microorganisms such as the bacterium Myxococcus xanthus. Here, we develop a model for spinning self-propelled rods on a substrate, and we coarse grain it to derive the corresponding hydrodynamic equations. If the rods propel purely along their axis, they form an active nematic at high density and activity. However, spinning rods can also roll sideways as they move. We find that this transverse motion turns the system into a chiral active nematic. Thus, we identify a mechanism whereby individual chirality can give rise to collective local chiral flows. Finally, we analyze experiments on M. xanthus colonies to show that they exhibit chiral flows around topological defects, with a chiral activity about an order of magnitude weaker than the achiral one. Our work reveals the collective behavior of active screws, which is relevant to colonies of social bacteria and groups of unicellular parasites.