According to quantum theory, measurement and backaction are inextricably linked. In optical position measurements, this backaction is known as radiation pressure shot noise. In analogy, a measurement of the orientation of a mechanical rotor must disturb its angular momentum by radiation torque shot noise. In this work, we observe the shot-noise torque fluctuations arising in a measurement of the angular orientation of an optically levitated nanodumbbell. We feedback cool the dumbbells rotational motion and investigate its reheating behavior when released from feedback. In high vacuum, the heating rate due to radiation torque shot noise dominates over the thermal and technical heating rates in the system.