Plasma rotation plays a critical role in improving plasma confinement in a magnetically confined fusion device. Spontaneous plasma rotation and its reversal of orientation without external momentum input have been observed in some tokamak fusion devices, while the underlying physics is not well understood. A new mechanism based on neoclassical toroidal plasma viscosity induced by symmetry breaking is proposed and well reproduces both spontaneous toroidal rotation and its reversals in tokamaks by considering a small non-axisymmetric displacement in the plasma center, since internal instabilities are often observed in those experiments. The nonlinear hysteresis process of plasma rotation reversals is also well reproduced in the modeling. The mechanism for driving spontaneous plasma rotation proposed in this paper may be utilized for achieving more economical operation of future tokamak fusion reactors.