We demonstrate that the transfer matrix of the inhomogeneous N-state chiral Potts model with two vertical superintegrable rapidities serves as the Q operator of the XXZ chain model for a cyclic representation of with Nth root-of-unity and representation parameter for odd N. The symmetry problem of the XXZ chain with a general cyclic representation is mapped onto the problem of studying the Q operator of some special one-parameter family of generalized τ⁽²⁾ models. In particular, the spin-(N−1)/2 XXZ chain model with and the homogeneous N-state chiral Potts model at a specific superintegrable point are unified as one physical theory. By Baxter's method, developed for producing a Q₇₂ operator of the root-of-unity eight-vertex model, we construct the Q_R,Q_L and Q operators of a superintegrable τ⁽²⁾ model, then identify them with transfer matrices of the N-state chiral Potts model for a positive integer N. We thus obtain a new method of producing the superintegrable N-state chiral Potts transfer matrix from the τ⁽²⁾ model by constructing its Q operator.