  The goal of the present paper is to push Sz.-Nagy--Foias model theory for a
completely nonunitary Hilbert-space contraction operator $T$, to the case of a
commuting pair of contraction operators $(T_1, T_2)$ having product $T = T_1
T_2$ which is completely nonunitary. The idea is to use the Sz.-Nagy-Foias
functional model for $T$ as the model space also for the commutative tuple
($T_1, T_2)$ with $T = T_1 T_2$ equal to the usual Sz.-Nagy--Foias model
operator, and identify what added structure is required to classify such
commutative contractive factorizations $T = T_1 T_2$ up to unitary equivalence.
In addition to the characteristic function $\Theta_T$, we identify additional
invariants $({\mathbb G}, {\mathbb W})$ which can be used to construct a
functional model for the commuting pair $(T_1, T_2)$ and which have good
uniqueness properties: if two commutative contractive pairs $(T_1, T_2)$ and
$(T'_1, T'_2)$ are unitarily equivalent, then their characteristic triples
$(\Theta, {\mathbb G}, {\mathbb W})_T$ and $(\Theta, {\mathbb G}, {\mathbb
W})_{T'}$ coincide in a natural sense. We illustrate the theory with several
simple cases where the characteristic triples can be explicitly computed. This
work extends earlier results of Berger-Coburn-Lebow \cite{B-C-L} for the case
where $(T_1, T_2)$ is a pair of commuting isometries, and of Das-Sarkar
\cite{D-S}, Das-Sarkar-Sarkar \cite{D-S-S} and the second author \cite{sauAndo}
for the case where $T = T_1T_2$ is pure (the operator sequence $T^{*n}$ tends
strongly to $0$). Finally we use the model to study the structure of joint
invariant subspaces for a commutative, contractive operator pair, extending
results of Sz.-Nagy--Foias for the single-operator case.
