An assembly for an aircraft propulsion system (20) includes a bladed rotor (24) and a thrust vectoring exhaust nozzle (100). The bladed rotor (24) is rotatable about an axis (28). The thrust vectoring exhaust nozzle (100) is configured to direct gas propelled by the bladed rotor (24) out of the aircraft propulsion system (20) along a first direction during a first mode and along a second direction during a second mode. The first direction is parallel with the axis (28) or angularly offset from the axis (28) by no more than five degrees. The second direction is angularly offset from the axis (28) by at least seventy-five degrees. The thrust vectoring exhaust nozzle (100) has a first exit area (109A) during the first mode and a second exit area (109B) during the second mode that is greater than the first exit area (109A).