We report on radio polarimetry observations for nine southern pulsars. Six of the nine in the sample are young, with characteristic ages under 100 kyr and high spin-down luminosities. All six show a significant degree of linear polarization. We also confirm a previously noticed trend in which the degree of linear polarization increases with spin-down luminosity. Where possible, we have used the rotating-vector model of the pulsar emission geometry to fit the observed position angle data. Our fit for PSR J1513-5908 (B1509-58) in particular is useful for directly testing the magnetospheric model of Melatos in combination with further timing observations. For this pulsar we find that a magnetic inclination angle greater than or equal to 60 degrees is excluded at the 3 sigma level and that the geometry suggested by the morphology of an apparent bipolar X-ray outflow is marginally inconsistent with the Melatos model. We also report on the polarimetry of three older pulsars: PSR J0045-7319, PSR J1627-4850, and PSR J1316-6232 (whose discovery we also report). Of these, only PSR J0045-7319 shows significant polarization.