It has been proved in [5] that if a graph $G$ is isomorphic to a cubic $(m, n)$-metacirculant graph $MC(m,n,\alpha,S_0, S_1,\dots,S_{\mu})$ with $S_0\not = \emptyset$, then $G$ is isomorphic to either a union of finitely many disjoint copies of a circulant graph $C(2\ell,S)$, where $\ell > 1$ and $S=\{1, -1, \ell\}$ or a union of finitely many disjoint copies of a generalized Petersen graph $GP(d,k)$, where $d > 2$ and $k^2\equiv \pm 1$ (mod $d$). In this paper, we prove that the converse is also true.