  In this paper we consider three submonoids of the dihedral inverse monoid
$\mathcal{DI}_n$, namely its submonoids $\mathcal{OPDI}_n$, $\mathcal{MDI}_n$
and $\mathcal{ODI}_n$ of all orientation-preserving, monotone and
order-preserving transformations, respectively. For each of these three
monoids, we compute the cardinal, give descriptions of Green's relations and
determine the rank.
