An asymmetric relativistic model for FRII radio sources is described which takes account of both relativistic effects and intrinsic/environmental asymmetries to explain the observed structural asymmetry of their radio lobes. A key feature of the model is jet-sidedness, which can now be determined for about 80% of the FRII sources in the 3CRR complete sample. It is shown that a simple asymmetric relativistic model can account for a wide range of observational data, and that the relativistic and intrinsic asymmetry effects are of comparable importance. The mean translational speed of the lobes is . The results are in agreement with an orientation-based unified scheme in which the critical angle separating the radio galaxies from the radio quasars is about 50°.