THRUST REVERSER SYSTEM WITH HIDDEN TURNING DOOR

An assembly is provided for an aircraft propulsion system. This assembly includes a thrust reverser system. The thrust reverser system includes a cavity, a sleeve and a turning door. The sleeve is configured to translate along a centerline between a sleeve stowed position and a sleeve deployed position. The turning door is configured to move between a turning door stowed position and a turning door deployed position. The turning door is disposed within the cavity when the sleeve is disposed in the sleeve stowed position. The turning door projects in a radial outward direction away from the sleeve and the centerline when the sleeve is in the sleeve deployed position.