  We obtain a complete characterization for doubly commuting mixed invariant
subspaces of the Hardy space over the unit polydisc. We say a closed subspace
$\clq$ of $H^2(\D^n)$ is a mixed invariant if $M_{z_{j}}(\clq) \subseteq \clq$
for $1 \leq j \leq k$ and $M_{z_{j}}^*(\clq) \subseteq \clq$, $k+1 \leq j \leq
n$ for some integer $k \in \{1, 2, \ldots, n-1 \}$. We prove that a mixed
invariant subspace $\clq$ of $H^2(\D^n)$ is doubly commuting if and only if \[
\clq = \Theta H^2(\D^k) \otimes \clq_{\theta_1} \otimes \cdots \otimes
\clq_{\theta_{n-k}}, \] where $\Theta \in H^{\infty}(\D^k)$ is some inner
function and $\clq_{\theta_j}$ is either a Jordan block $H^2(\D)\ominus
\theta_j H^2(\D)$ for some inner function $\theta_j$ or the Hardy space
$H^2(\D)$. Furthermore, an explicit representation of commutant of n-tuple
doubly commuting shifts as well as a representation of commutant of doubly
commuting shifts and adjoint of shifts are obtained. Finally, we discuss some
concrete examples of mixed invariant subspaces.
