We present a proprioceptive teleoperation system that uses a reflexive grasping algorithm to enhance the speed and robustness of pick-and-place tasks. The system consists of two manipulators that use quasi-direct-drive actuation to provide highly transparent force feedback. The end-effector has bimodal force sensors that measure 3-axis force information and 2-dimensional contact location. This information is used for anti-slip and re-grasping reflexes. When the user makes contact with the desired object, the re-grasping reflex aligns the gripper fingers with antipodal points on the object to maximize the grasp stability. The reflex takes only 150ms to correct for inaccurate grasps chosen by the user, so the user's motion is only minimally disturbed by the execution of the re-grasp. Once antipodal contact is established, the anti-slip reflex ensures that the gripper applies enough normal force to prevent the object from slipping out of the grasp. The combination of proprioceptive manipulators and reflexive grasping allows the user to complete teleoperated tasks with precision at high speed.