We study coadjoint $B$-orbits on $\mathfrak{n}^*$, where $B$ is a Borel subgroup of a complex orthogonal group $G$, and $\mathfrak{n}$ is the Lie algebra of the unipotent radical of $B$. To each basis involution $w$ in the Weyl group $W$ of $G$ one can assign the associated $B$-orbit $Ω_w$. We prove that, given basis involutions $σ$, $τ$ in $W$, if the orbit $Ω_σ$ is contained in the closure of the orbit $Ω_τ$ then $σ$ is less than or equal to $τ$ with respect to the Bruhat order on $W$. For a basis involution $w$, we also compute the dimension of $Ω_w$ and present a conjectural description of the closure of $Ω_w$.