Detailed studies of the non-linear radial wavenumber spectrum of electric potential fluctuations in gyro-kinetic plasma turbulence simulations have led to a new paradigm that is capable of computing the momentum pinch quasilinearly. It is found that shear in the E × B velocity Doppler shift suppresses turbulence by inducing a shift in the peak of the radial wavenumber spectrum, and a reduction in the amplitude. An analytic model of the process is used to understand the roles of the sheared velocity and the non-linear mode coupling. The analytic model leads to a simple formula that fits the non-linear spectrum and only depends on the spectral average shift in the radial wavenumber. This 'spectral shift' model is a new paradigm that radial parity breaking is the fundamental mechanism that suppresses the turbulence through a radial wavenumber shift. The E × B velocity shear is one of a number of radial parity breaking mechanisms. Using a model of the spectral shift the toroidal Reynolds stress due to the E × B velocity shear can be computed for the first time with a quasilinear model. It is shown that, when diamagnetic and neoclassical contributions to the parallel flows are included, the E × B velocity shear term in the toroidal Reynolds stress allows the sign of the intrinsic toroidal rotation to change. Simulations of the co-current and balanced neutral beam injection phase of a DIII-D discharge using the quasilinear model show good agreement with experiment.