A thrust reverser for an aircraft propulsion assembly, this reverser including a lower door and an upper door defining, in thrust reversal configuration, a deflection opening, through which a portion of the fluid not serving to produce the thrust reversal of the aircraft can exit the reverser downstream. The downstream edge of the lower door is offset towards the rear with respect to the downstream edge of the upper door so as to orient the fluid flow passing through the deflection opening vertically upwards. When the propulsion assembly is mounted at the rear portion of the fuselage of the aircraft, this makes it possible in particular to improve the supply of the control surface of the aircraft in the landing phase, in particular in crosswind conditions.