A thrust reverser for an aircraft propulsion assembly includes a movable structure provided with external covers connected to hinges through a pivot connection allowing the external covers to rotate between a closed position and a maintenance position. The hinges are connected to secondary rails rigidly attached to an engine pylon through a sliding connection allowing the external covers to move in translation between a direct thrust position and a reverse thrust position. The hinges and the secondary rails are independent of beams to which the parts of the movable structure other than the external covers are connected. A method for maintaining an aircraft propulsion unit is also disclosed. The method includes positioning of the external covers in maintenance position followed by a placement of the beams and of the parts of the movable structure other than the external covers.