Two types of supernova are thought to produce the overwhelming majority of neutron stars in the Universe(1). The first type, iron-core-collapse supernovae, occurs when a high-mass star develops a degenerate iron core that exceeds the Chandrasekhar limit(2). The second type, electron-capture supernovae, is associated with the collapse of a lower-mass oxygen-neon-magnesium core as it loses pressure support owing to the sudden capture of electrons by neon and/or magnesium nuclei(3,4). It has hitherto been impossible to identify the two distinct families of neutron stars produced in these formation channels. Here we report that a large, well-known class of neutron-star-hosting X-ray pulsars is actually composed of two distinct subpopulations with different characteristic spin periods, orbital periods and orbital eccentricities. This class, the Be/X-ray binaries, contains neutron stars that accrete material from a more massive companion star(5). The two subpopulations are most probably associated with the two distinct types of neutron-star-forming supernova, with electron-capture supernovae preferentially producing systems with short spin periods, short orbital periods and low eccentricities. Intriguingly, the split between the two subpopulations is clearest in the distribution of the logarithm of spin period, a result that had not been predicted and which still remains to be explained.