We develop a theory for the current-direction dependence of the normal-zone propagation velocity observed in multifilamentary composite superconductors. Normal-zone propagation is known to be driven primarily by Joule heating in both the normal zone and the current-sharing zone, which lies between the normal (T>Tc) and superconducting (I<Ic) zones. We show, however, that the Peltier effect is an important secondary source of heating and cooling in the current-sharing zone and accounts for essentially all of the current-direction dependence of the normal-zone propagation velocity observed in a Nb3Sn/Cu composite conductor. For one current direction Peltier heat is deposited in the current-sharing zone, thereby adding to the Joule heat and increasing the normal-zone propagation velocity, while for the other current direction Peltier heat is absorbed, thereby cooling the current-sharing zone and thus decreasing the normal-zone propagation velocity. Contrary to the claim of Gurevich and Mints, we find that the Thomson effect is about an order of magnitude too small to account for the current-direction dependence observed in a Nb3Sn/Cu composite conductor.