  In this paper we analyze and classify the totally geodesic subspaces of
finite volume quaternionic hyperbolic orbifolds and their generalizations,
locally symmetric orbifolds arising from irreducible lattices in Lie groups of
the form $(\mathbf{Sp}_{2n}(\mathbb{R}))^q \times \prod_{i=1}^r
\mathbf{Sp}(p_i,n-p_i) \times (\mathbf{Sp}_{2n}(\mathbb{C}))^s$. We give
criteria for when the totally geodesic subspaces of such an orbifold determine
its commensurability class. We give a parametrization of the commensurability
classes of finite volume quaternionic hyperbolic orbifolds in terms of
arithmetic data, which we use to show that the complex hyperbolic totally
geodesic subspaces of a quaternionic hyperbolic orbifold determine its
commensurability class, but the real hyperbolic totally geodesic subspaces do
not. Lastly, our tools allow us to show that every cocompact lattice
$\Gamma<\mathbf{Sp}(m,1)$, $m\ge 2$, contains quasiconvex surface subgroups.
