The control of plasma rotation in a tokamak by controlling the impurity content is investigated. A neoclassical theory for momentum transport by collisional ions in a tokamak plasma with strong neutral beam injection and strong rotation is developed. A consistently ordered hierarchy of approximations to the kinetic equation are derived and solved to obtain expressions for particle flows, the radial electric field, poloidal asymmetries in density and potential, and the radial flux of toroidal angular momentum and the associated torque that acts to damp toroidal rotation. Upon decomposing the first-order distribution function into gyroangle-dependent and gyroangle-averaged components, neoclassical gyroviscosity is recovered from the former, and a new “rotational” viscosity of a collisional origin is recovered from the latter. The same viscosity coefficient and functional form are obtained for both types of viscosity.