  In this paper we consider the submonoids $\mathcal{OPDI}_n$,
$\mathcal{MDI}_n$ and $\mathcal{ODI}_n$ of the dihedral inverse monoid
$\mathcal{DI}_n$ of all orientation-preserving, monotone and order-preserving
transformations, respectively. Our goal is to exhibit presentations for each of
these three monoids.
