Scaling of intrinsic rotation in DIII-D H-mode plasmas demonstrates a strong correlation with the ion temperature (Ti) and stored plasma thermal energy, indicating a coupling between the turbulent intrinsic momentum flux and the turbulent energy flux. We consider intrinsic rotation to be the toroidal rotation in axisymmetric conditions with no external momentum injection. The DIII-D dimensionless empirical scaling of intrinsic rotation with plasma stored energy has been recently tested by novel experiments on DIII-D that utilize relatively small variations in the plasma shape, namely the triangularity, to modify the intrinsic rotation. Shape variation affects the intrinsic rotation by modifying the turbulent transport, rather than via changes in the auxiliary heating power, or applied torque. These H-modes are heated by ECH with no external torque input. Balanced torque blips from neutral beams measure the ion flow velocity and Ti. Higher thermal energy and intrinsic angular momentum are correlated with higher triangularity. Turbulent density fluctuations in the pedestal region show a significantly higher level at lower triangularity, with lower energy confinement, possibly the source of greater transport. In DIII-D, the E × B shear, which is mainly driven by the edge pressure gradient term in intrinsic rotation conditions, could provide the dominant symmetry breaking necessary for generating a net turbulent momentum stress and qualitatively agree with the scaling.