  Let $\mathcal{X}_S$ denote the class of spaces homeomorphic to two closed
orientable surfaces of genus greater than one identified to each other along an
essential simple closed curve in each surface. Let $\mathcal{C}_S$ denote the
set of fundamental groups of spaces in $\mathcal{X}_S$. In this paper, we
characterize the abstract commensurability classes within $\mathcal{C}_S$ in
terms of the ratio of the Euler characteristic of the surfaces identified and
the topological type of the curves identified. We prove that all groups in
$\mathcal{C}_S$ are quasi-isometric by exhibiting a bilipschitz map between the
universal covers of two spaces in $\mathcal{X}_S$. In particular, we prove that
the universal covers of any two such spaces may be realized as isomorphic cell
complexes with finitely many isometry types of hyperbolic polygons as cells. We
analyze the abstract commensurability classes within $\mathcal{C}_S$: we
characterize which classes contain a maximal element within $\mathcal{C}_S$; we
prove each abstract commensurability class contains a right-angled Coxeter
group; and, we construct a common CAT$(0)$ cubical model geometry for each
abstract commensurability class.
