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This paper presents a novel family of fully isotropic parallel mechanisms whose output link is provided with Schoenflies motion, i.e., it can freely translate in space and rotate about a fixed direction. A methodology is proposed that makes use of the theory of screws to synthesize desired forms for both the direct and the inverse Jacobian matrices. In particular, these are made diagonal and constant throughout the workspace. Motors are mounted one per leg and each one of them actuates one of the degrees of freedom of the output body through a constant one-to-one velocity relation. As a consequence, motors may apply, with equal ease, a twist or a wrench of any amplitude (within the motor operation range) to the end-effector around any screw congruous with the admitted motion, so that full isotropy is achieved. Kinematic analysis is trivial and no computation is required for real-time control. Furthermore, actuator motion range