From a natural generalization to $Z^2$ of the concept of congruence, it is possible to define a family of $2$-regular digraphs that we call "commutative-step networks". Particular examples of such digraphs are the Cartesian product of two directed cycles, $C_l\times C_h$, and the "fixed-step network" (or "$2$-step circulant digraph") $D_{N,a,b}$. In this paper the theory of congruence in $Z^2$ is applied to derive three equivalent characterizations of those commutative-step networks that have a Hamiltonian cycle. Some known results are then obtained as a corollary. For instance, necessary and sufficient conditions for C_l\times C_h$ or $D_{N,a,b}$ to be Hamiltonian are discussed.

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