  Let $S_{E}$ be the shift operator on vector-valued Hardy space $H_{E}^{2}.$
Beurling-Lax-Halmos Theorem identifies the invariant subspaces of $S_{E}$ and
hence also the invariant subspaces of the backward shift $S_{E}^{\ast}.$ In
this paper, we study the invariant subspaces of $S_{E}\oplus S_{F}^{\ast}.$ We
establish a one-to-one correspondence between the invariant subspaces of
$S_{E}\oplus S_{F}^{\ast}$ and a class of invariant subspaces of bilateral
shift $B_{E}\oplus B_{F}$ which were described by Helson and Lowdenslager. As
applications, we express invariant subspaces of $S_{E}\oplus S_{F}^{\ast}$ as
kernels or ranges of mixed Toeplitz operators and Hankel operators with partial
isometry-valued symbols. Our approach greatly extends and gives different
proofs of the results of C\^{a}mara and Ross, and Timotin where the case with
one dimensional $E$ and $F$ was considered.
