The turbulent convective flux of the toroidal angular momentum density is derived using the nonlinear toroidal gyrokinetic equation which conserves phase space density and energy [T. S. Hahm, Phys. Fluids, 31, 2670 (1988)]. A novel pinch mechanism is identified which originates from the symmetry breaking due to the magnetic field curvature. A net parallel momentum transfer from the waves to the ion guiding centers is possible when the fluctuation intensity varies on the flux surface, resulting in imperfect cancellation of the curvature drift contribution to the parallel acceleration. This mechanism is inherently a toroidal effect, and complements the k∥ symmetry breaking mechanism due to the mean E×B shear [O. Gurcan, Phys. Plasmas 14, 042306 (2007)] which exists in a simpler geometry. In the absence of ion thermal effects, this pinch velocity of the angular momentum density can also be understood as a manifestation of a tendency to homogenize the profile of "magnetically weighted angular momentum density," n mi R2 ω∥ B2. This part of the pinch flux is mode-independent (whether it is trapped electron mode or ion temperature gradient mode driven), and radially inward for fluctuations peaked at the low- B -field side, with a pinch velocity typically, V Ang TEP ∼-2 φ R0. Ion thermal effects introduce an additional radial pinch flux from the coupling with the curvature and grad- B drifts. This curvature driven thermal pinch can be inward or outward, depending on the mode-propagation direction. Explicit formulas in general toroidal geometry are presented. © 2007 American Institute of Physics.