ABSTRACT We show that ultraluminous X-ray sources (ULXs) with coherent X-ray pulsing (PULXs) probably have neutron-star spin axes significantly misaligned from their central accretion discs. Scattering in the funnels collimating their emission and producing their apparent super-Eddington luminosities is the most likely origin of the observed correlation between pulse fraction and X-ray photon energy. Pulsing is suppressed in systems with the neutron-star spin closely aligned to the inner disc, explaining why some ULXs show cyclotron features indicating strong magnetic fields, but do not pulse. We suggest that alignment (or conceivably, field suppression through accretion) generally occurs within a fairly short fraction of the ULX lifetime, so that most neutron-star ULXs become unpulsed. As a result we further suggest that almost all ULXs actually have neutron-star accretors, rather than black holes or white dwarfs, reflecting their progenitor high-mass X-ray binary and supersoft X-ray source populations.