  Let $(\mathsf{hypo}_n,~^\sharp)$ be the hypoplactic monoid of finite rank $n$
with Sch\"{u}tzenberger's involution $^{\sharp}$. In this paper, we exhibit a
faithful representation of $(\mathsf{hypo}_n,~^\sharp)$ as an involution monoid
of upper triangular matrices over any semiring from a large class including the
tropical semiring under the skew transposition. We then give a transparent
combinatorial characterization of the word identities satisfied by
$(\mathsf{hypo}_n,~^\sharp)$. Further, we prove that
$(\mathsf{hypo}_n,~^\sharp)$ is non-finitely based if and only if $n=2, 3$ and
give a polynomial time algorithm to check whether a given word identity holds
in $(\mathsf{hypo}_n,~^\sharp)$.
