We extend the theory of shear suppression of turbulence in the toroidal geometry (1995 Phys. Plasmas2 1648) by considering initially tilted turbulence eddies which are routinely observed in nonlinear simulations of tokamak turbulence. Using the two point nonlinear decorrelation theory, we obtain a new shear suppression criterion which depends on the relative sign of shear and initial tilting angle. Our result implies a different level of turbulent transport depending on this relative sign.