The behaviour of a dispersion of infinitely polarizable slender rods in an electric field is described using theory and numerical simulations. The polarization of the rods results in the formation of dipolar charge clouds around the particle surfaces, which in turn drive nonlinear electrophoretic flows under the action of the applied field. This induced-charge electrophoresis causes no net migration for uncharged particles with fore-aft symmetry, but can lead to rotations and to relative motions as a result of hydrodynamic interactions. A slender-body formulation is derived that accounts for induced-charge electrophoresis based on a thin double layer approximation, and shows that the effects of the electric field on a single rod can be modelled by a linear slip velocity along the rod axis, which causes particle alignment and drives a stresslet flow in the surrounding fluid. Based on this slender-body model, the hydrodynamic interactions between a pair of aligned rods are studied, and we identify domains of attraction and repulsion, which suggest that particle pairing may occur. An efficient method is implemented for the simulation of dispersions of many Brownian rods undergoing induced-charge electrophoresis, that accounts for far-field hydrodynamic interactions up to the stresslet term, as well as near-field lubrication and contact forces. Simulations with negligible Brownian motion show that particle pairing indeed occurs in the suspension, as demonstrated by sharp peaks in the pair distribution function. The superposition of all the electrophoretic flows driven on the rod surfaces is observed to result in a diffusive motion at long times, and hydrodynamic dispersion coefficients are calculated. Results are also presented for colloidal suspensions, in which Brownian fluctuations are found to hinder particle pairing and alignment. Orientation distributions are obtained for various electric field strengths, and are compared to an analytical solution of the Fokker-Planck equation for the orientation probabilities in the limit of infinite dilution.