There is substantial observational evidence against the symmetric relativistic model of FR II radio sources. An asymmetric relativistic model is proposed which takes account of both relativistic effects and intrinsic/environmental asymmetries to explain the structural asymmetries of their radio lobes. A key parameter of the model is the jet-side of the double sources, which is estimated for 80 per cent of the FR II sources in the 3CRR complete sample. Statistical analyses of the properties of these sources show that the asymmetric model is in agreement with a wide range of observational data, and that the relativistic and intrinsic asymmetry effects are of comparable importance. Intrinsic/environmental asymmetry effects are more important at high radio luminosities and small physical scales. The mean translational speed of the lobes is found to be consistent with the speeds found from spectral ageing arguments. According to a Gaussian model, the standard deviation of the distribution of vlobe is σvl = 0.04c. The results are in agreement with an orientation-based unification scheme in which the critical angle separating the radio galaxies from quasars is about 45°.