Let \({\mathcal M}\) be a non-trivial shift-invariant subspace of \(H^ 2(T^ 2)\). It is known [cf. W. Rudin, Function theory in polydiscs (1974; Zbl 0294.32001)] that in general \({\mathcal M}\) is not of the form \(qH^ 2(T^ 2)\) for any inner function q. The author’s main result is that \({\mathcal M}\) is of the form \(q\cdot H^ 2\) if and only if the operators \(V_ 1\) and \(V_ 2\) are doubly commuting on \({\mathcal M}\). Here, \(V_ i\) is multiplication on \(H^ 2(T^ 2)\) by \(t_ i\) where \(t=(t_ 1,t_ 2)\in T^ 2\), and doubly commuting means that \(V_ 1\) commutes with both \(V_ 2\) and \(V^*_ 2\). As a consequence, all invariant subspaces of the form \(q\cdot H^ 2\) for some inner function q are unitarily equivalent. In addition, if \(f\in H^ 2(T^ 2)\), then \({\mathcal M}_ f=\overline{span}\{V^ m_ 1V^ n_ 2f:\) m,n\(\geq 0\}\) is of the form \(q\cdot H^ 2\) if and only if \(V_ 1\) and \(V_ 2\) are doubly commuting on \({\mathcal M}_ f\). The proof of the main theorem uses a result of M. Slocinski [Ann. Pol. Math. 37, 255-262 (1980; Zbl 0485.47018)] on commuting isometries. In addition, the relation of this work to a recent paper of O. P. Agrawal, D. N. Clark and R. G. Douglas [Pac. J. Math. 121, 1-11 (1986; Zbl 0609.47012)] on unitary equivalence of invariant subspaces of \(H^ 2(T^ n)\) is discussed. The results in these papers are essentially disjoint.