It is shown that, if one considers the “total freedom” of instantaneous motion, three distinct types of screw systems arise. Of these only the ones which maintain the kinematic constraints—the so-called reciprocal screw systems—have been studied before. This paper introduces and studies the two new types, which we name “repelling” and “contrary” screws. We present a detailed theoretical study of the repelling screw systems, these as well as the other screw systems are useful in the mathematical modelling of the parts assembly process and the study of direct contact mechanisms.