ABSTRACT In the canonical picture of pulsars, radio emission arises from a narrow cone centred on the star’s magnetic axis but many basic details remain unclear. We use high-quality polarization data taken with the Parkes radio telescope to constrain the geometry and emission heights of pulsars showing interpulse emission, and include the possibility that emission heights in the main and interpulse may be different. We show that emission heights are low in the centre of the beam, typically less than 3 per cent of the light cylinder radius. The emission beams are underfilled in longitude, with an average profile width only 60 per cent of the maximal beamwidth and there is a strong preference for the visible emission to be located on the trailing part of the beam. We show substantial evidence that the emission heights are larger at the beam edges than in the beam centre. There is some indication that a fan-like emission beam explains the data better than conal structures. Finally, there is a strong correlation between handedness of circular polarization in the main and interpulse profiles, which implies that the hand of circular polarization is determined by the hemisphere of the visible emission.