This paper deals with the problem of classifying linear \(N\)-periodic systems of difference equations. Such systems have the form: \[ x_{n +1}=A_nx_n,\;n\in\mathbb{Z} A_n: \mathbb{C}^{r_n} \to\mathbb{C}^{r_{n+1}},\tag{1} \] with \(A_n=A_{n+N}\) for \(n\in\mathbb{Z}\). The coefficient \(A_n\) in (1) is assumed to be a linear transformation and, by \(N\)-periodicity, \(r_{n+N}=r_n\) for each \(n\). The authors’ goal is to classify \(N\)-periodic difference equations under two types of time-variant basis transformations: kinematic similarity and \(N\)-periodic similarity. Here the authors give necessary and sufficient conditions for two \(N\)-periodic difference equations to be kinematically similar or \(N\)-periodically similar.