Polarimetric studies of pulsar radio emission traditionally concentrate on how the Stokes vector (I, Q, U, V) varies with pulse longitude, with special emphasis on the position angle (PA) swing of the linearly polarized component. The interpretation of the PA swing in terms of the rotating vector model is limited by the assumption of an axisymmetric magnetic field and the degeneracy of the output with respect to the orientation and magnetic geometry of the pulsar; different combinations of the latter two properties can produce similar PA swings. This paper introduces Stokes phase portraits as a supplementary diagnostic tool with which the orientation and magnetic geometry can be inferred more accurately. The Stokes phase portraits feature unique patterns in the I-Q, I-U, and Q-U planes, whose shapes depend sensitively on the magnetic geometry, inclination angle, beam and polarization patterns, and emission altitude. We construct look-up tables of Stokes phase portraits and PA swings for pure and current-modified dipole fields, filled core and hollow cone beams, and two empirical linear polarization models, L/I = \cos \theta_0 and L/I = \sin \theta_0, where \theta_0 is the colatitude of the emission point. We compare our look-up tables to the measured phase portraits of 24 pulsars in the European Pulsar Network online database. We find evidence in 60% of the objects that the radio emission region may depart significantly from low altitudes, even when the PA swing is S-shaped and/or the pulse-width-period relation is well satisfied. On the other hand, the data are explained adequately if the emission altitude exceeds ~10% of the light cylinder radius. We conclude that Stokes phase portraits should be analysed concurrently with the PA swing and pulse profiles in future when interpreting radio pulsar polarization data. ; Comment: 60 pages, 58 figures, submitted to MNRAS, accepted 13 Oct 2010