  A pair of Hilbert space linear operators $(V_1,V_2)$ is said to be
$q$-commutative, for a unimodular complex number $q$, if $V_1V_2=qV_2V_1$. A
concrete functional model for $q$-commutative pairs of isometries is obtained.
The functional model is parametrized by a collection of Hilbert spaces and
operators acting on them. As a consequence, the collection serves as a complete
unitary invariance for $q$-commutative pairs of isometries. A $q$-commutative
operator pair $(V_1,V_2)$ is said to be doubly $q$-commutative, if in addition,
it satisfies $V_2V_1^*=qV_1^*V_2$. Doubly $q$-commutative pairs of isometries
are also characterized. Special attention is given to doubly $q$-commutative
pairs of shift operators. The notion of $q$-commutativity is then naturally
extended to the case of general tuples of operators to obtain a similar model
for tuples of $q$-commutative isometries.
