There is substantial observational evidence against the symmetric relativistic model of FRII 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 % of the FRII 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 \overline { v } _ { lobe } = ( 0.11 \pm 0.013 ) c , consistent with the speeds found from spectral ageing arguments .
According to a Gaussian model , the standard deviation of the distribution of v _ { lobe } is \sigma _ { v _ { l } } = 0.04 c .
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 ^ { \circ } .