This article presents a back-drivable spherical wrist with smart compliance for rendering human wrist-like capabilities in robotic applications. In order to synthetically impose both motion accuracy and active compliance for dedicated wrist manipulations, a robust hybrid control method is proposed that can switch between motion control (at free motion) and impedance control (at interactions with the environment). A sliding manifold is formulated so that when the control law drives the sliding manifold to zero, the motion/impedance errors will automatically converge to bounded regions despite system uncertainties/inaccuracies. Due to the direct-drive configuration of the proposed spherical wrist, the contact torque can be precisely estimated with an observer, providing an intuitive criterion of contact based on which the control modes can switch autonomously without external force/torque sensors and predefined geometric boundaries that are usually required in conventional hybrid force/position control. The concept is validated on a spherical wrist test-bed in the context of a throat swabbing process. The performance of the hybrid motion-impedance control evaluated against conventional impedance and admittance control methods demonstrate that the proposed spherical wrist enables a robotic system with both fine manipulations and smooth compliant interactions in wrist space.