Radio pulsars are thought to born with spin periods of 0.02–0.5 s and space velocities of 100–1,000 kms^(-1), and they are inferred to have initial dipole magnetic fields of 10^(11)–10^(13) G. The average space velocity of their progenitor stars is less than 15 kms^(-1), which means that pulsars must receive a substantial ‘kick’ at birth. Here we propose that the birth characteristics of pulsars have a simple physical connection with each other. Magnetic fields maintained by differential rotation between the core and envelope of the progenitor would keep the whole star in a state of approximately uniform rotation until 10 years before the explosion. Such a slowly rotating core has 1,000 times less angular momentum than required to explain the rotation of pulsars. The specific physical process that ‘kicks’ the neutron star at birth has not been identified, but unless its force is exerted exactly head-on it will also cause the neutron star to rotate. We identify this process as the origin of the spin of pulsars. Such kicks may cause a correlation between the velocity and spin vectors of pulsars. We predict that many neutron stars are born with periods longer than 2 s, and never become radio pulsars.