Let X be an irreducible algebraic curve defined over a finite field Fq of characteristic p > 2.Assume that the Fq-automorphism group of X admits as an automorphism group the direct product of two cyclic groups Cm and Cn of orders m and n prime to p such that both quotient curves X /Cn and X /Cm are rational. In this paper, we provide a complete classification of such curves, as well as a characterization of their full automorphism groups.