  Let $(\mathsf{baxt}_n,~^\sharp)$ be the Baxter monoid of finite rank $n$ with
Sch\"{u}tzenberger's involution $^{\sharp}$. In this paper, it is shown that
$(\mathsf{baxt}_n,~^\sharp)$ admits a faithful representation by an involution
monoid of upper triangular matrices over any semiring from a large class
including the tropical semiring under the skew transposition. Then a
transparent combinatorial characterization of the word identities satisfied by
$(\mathsf{baxt}_n,~^\sharp)$ is given. Further, it is proved that
$(\mathsf{baxt}_n,~^\sharp)$ is finitely based if and only if $n\neq 3$, and
shown that the identity checking problem for $(\mathsf{baxt}_n,~^\sharp)$ can
be done in polynomial time.
