The internal rotation and magnetism of massive stars are considered in response to (i) the inward pumping of angular momentum through deep and slowly rotating convective layers, and (ii) the winding up of a helical magnetic field in radiative layers. Field winding can transport angular momentum effectively even when the toroidal field is limited by kinking. Magnetic helicity is pumped into a growing radiative layer from an adjacent convective envelope (or core). The receding convective envelope that forms during the early accretion phase of a massive star is the dominant source of helicity in its core, yielding a 1012 - 1013 G polar magnetic field in a collapsed neutron star (NS) remnant. Using MESA models of various masses, we find that the NS rotation varies significantly, from PNS ∼ 0.1-1 s in a 13 M☉ model to PNS ∼ 2 ms in a 25 M☉ model with an extended convective core. Stronger inward pumping of angular momentum is found in more massive stars, due to the growing thickness of the convective shells that form during the later stages of thermonuclear burning. On the other hand, stars that lose enough mass to form blue supergiants in isolation end up as very slow rotators. The tidal spin-up of a 40 M☉ star by a massive binary companion is found to dramatically increase the spin of the remnant black hole, allowing a rotationally supported torus to form during the collapse. The implications for post-collapse decay or amplification of the magnetic field are also considered.