The Chebyshev–Grübler–Kutzbach (CGK) mobility criterion is a very well known means of computing the relative degree of mobility possessed by a kinematic chain consisting of jointed rigid links. The relative degree of mobility indicates the number of independent parameters, such as revolute joint angles or prismatic joint translation distances, that are required to define a geometric configuration of the kinematic chain. In the 1800’s Pafnuty Chebyshev, Martin Grübler, and later in the early 1900’s Karl Kutzbach proposed mobility criteria that sum the unconstrained degree of freedom of the rigid links in the mechanical system and subtract the constraints imposed by the joints. Sometimes the CGK criterion yields an incorrect result. In this paper, the idea of accounting for the symmetry operations in the finite symmetry groups of the paradoxical kinematic chains that cause the criterion to predict degrees of mobility that contradict physical reality is proposed for the first time.