We generalize the Lagrangian transport theory to include the normal and shear Reynolds stresses, so that a complete tensor can be constructed. The ideology is based on imposition of the momentum and energy balance to a control volume moving at the local mean velocity, which bears the effect of de-coupling the mean from the fluctuation components. The resulting transport equations are verified, with available DNS data. Representation of the fluxes in this form leads to the dissipation scaling, which collapses the u′ 2, v′ 2 and u′v′ gradient profiles for all Reynolds numbers. Addition of the energy spectra, derivable from the maximum entropy principle, completes the Reynolds stress theory, to fully prescribe the turbulence structures in canonical geometries.