The present paper is continuation of \textit{K.~Izuchi} and \textit{T.~Nakazi} [Hokkaido Math.\ J.\ 33, 247--254 (2004; Zbl 1055.47008)] and \textit{K.~Izuchi, T.~Nakazi} and \textit{M.~Seto} [J.\ Operator Theory 51, 361--376 (2004; Zbl 1055.47009)]. Let \(\Gamma^2\) be the 2-dimensional unit torus and denote by \((z,w) = (e^{i\theta}, e^{i\theta})\) the variables in \(\Gamma^2\). Let further \(L^2 = L^2(\Gamma^2)\) denote the Hilbert space on \(\Gamma^2\) with the usual inner product \[ \langle f , g \rangle = \int_{\Gamma 2} f(e^{i\theta}, e^{i\theta})\bar g(e^{i\theta}, e^{i\theta})\,d\theta \,d\varphi /(2\pi)^2. \] The Fourier coefficients of \(f\in L^2\) are given by \[ f\,\hat{\,}(n, m) =\langle f , z^nw^m \rangle =\int_{\Gamma 2}f(e^{i\theta} , e^{i\theta})e^{-in\theta} e^{-im\varphi}\,d\theta \,d\varphi/(2\pi)^2. \] Let \(H^2\) denote the Hardy space on \(\Gamma^2\), that is, \[ H^2 = \{ f\in L^2\mid f\,\hat{\,}(n, m) = 0,\quad \text{if}\quad n < 0\;\text{or}\;m < 0 \}. \] For a bounded measurable function \(\psi\) on \(\Gamma^2\), let \(L_\psi\) denote the operator on \(L^2\) defined by \(L_\psi f = \psi f\) for \(f\in L^2\). Let \(N\) be a closed subspace of \(H^2\). Then the operator \(S_\psi\) on \(N\) is defined by \(S_\psi f =P_N L_\psi f\) for \(f\in N\), where \(P_N\) is the orthogonal projection from \(L^2\) onto \(N\). A closed subspace \(N\) of \(H^2\) is said to be backward shift invariant if \(zH^2\ni N\subset H^2 \ni N\), \(wH^2\ni N\subset H^2\ni wN\). In the paper under review, the authors give a characterization of backward shift invariant subspaces satisfying \(S_z^2 S_w^*= S_w^*S_z^2\).