  Let $U$ be a Sylow $p$-subgroup in a classical group over a finite field of
characteristic $p$. The coadjoint orbits of the group $U$ play the key role in
the description of irreducible complex characters of $U$. Almost all important
classes of orbits and characters studied to the moment can be uniformly
described as the orbits and characters associated with so-called orthogonal
rook placements. In the paper, we study such orbits for the orthogonal group.
We construct a polarization for the canonical form on such an orbit and present
a semi-direct decomposition for the corresponding irreducible characters in the
spirit of the Mackey little group method. As a corollary, we compute the
dimension of an orbit associated with an orthogonal rook placement.
