Natal kicks and spins are characteristic properties of neutron stars (NSs) and black holes (BHs). Both offer valuable clues to dynamical processes during stellar core collapse and explosion. Moreover, they influence the evolution of stellar multiple systems and the gravitational-wave signals from their inspiral and merger. Observational evidence of possibly generic spin-kick alignment has been interpreted as indication that NS spins are either induced with the NS kicks or inherited from progenitor rotation, which thus might play a dynamically important role during stellar collapse. Current three-dimensional supernova simulations suggest that NS kicks are transferred in the first seconds of the explosion, mainly by anisotropic mass ejection and, on a secondary level, anisotropic neutrino emission. In contrast, the NS spins are only determined minutes to hours later by angular momentum associated with fallback of matter that does not become gravitationally unbound in the supernova. Here, we propose a novel scenario to explain spin-kick alignment as a consequence of tangential vortex flows in the fallback matter that is accreted mostly from the direction of the NS's motion. For this effect the initial NS kick is crucial, because it produces a growing offset of the NS away from the explosion center, thus promoting onesided accretion. In this new scenario conclusions based on traditional concepts are reversed. For example, pre-kick NS spins are not required, and rapid progenitor-core rotation can hamper spinkick alignment. We also discuss implications for natal BH kicks and the possibility of tossing the BH's spin axis during its formation.