The present invention relates to a thrust reverser for the nacelle of a turbojet engine comprising, on the one hand, means (11) for deflecting at least some of an air flow of the turbojet engine and, on the other hand, at least one hood (10) able to move translationally in a direction substantially parallel to a longitudinal axis of the nacelle and able to switch alternately from a closed position in which it ensures the aerodynamic continuity of the nacelle and covers the deflection means, and an open position in which it opens a passage in the nacelle and uncovers the deflection means, characterized in that the moving hood comprises at least one outer part (10a) having a downstream extension forming a nozzle and at least one internal part (10b) each of which parts is mounted such that it is translationally mobile and is connected to at least one actuating means able to allow it to be moved, each independently of the other, or together, in a substantially longitudinal direction of the nacelle. The present invention also relates to a turbojet engine nacelle comprising such a thrust reverser.