Application of lower hybrid range of frequencies (LHRF) waves can induce both co- and counter-current directed changes in toroidal rotation in Alcator C-Mod plasmas, depending on the target plasma current, electron density, confinement regime and magnetic shear. For ohmic L-mode discharges with good core LH wave absorption, and significant current drive at a fixed LH power near 0.8 MW, the interior (r/a < 0.5) rotation increments (on a time scale of order the current relaxation time) in the counter-current direction if ne(10^20 m−3) > q95/11.5, and in the co-current direction if ne(10^20 m−3) < q95/11.5. All discharges with co-current rotation changes have q0 > 1, indicating a good correlation with driven current fraction, unifying the results observed on various tokamaks. For high density (ne >= 1.2 × 10^20 m−3) L-mode target discharges, where core LH wave absorption is low, the rotation change is in the co-current direction, but evolves on a shorter momentum transport time scale, and is seen across the entire spatial profile. For H-mode target plasmas, both co- and counter-current direction increments have been observed with LHRF. The H-mode co-rotation is correlated with the pedestal temperature gradient, which itself is enhanced by the LH waves absorbed in the plasma periphery. The H-mode counter-rotation increment, a flattening of the peaked velocity profile in the core, is consistent with a reduction in the momentum pinch correlated with a steepening of the core density profile. Most of these rotation changes must be due to indirect transport effects of LH waves on various parameters, which modify the momentum flux.