  The superspace ring $\Omega_n$ is a rank $n$ polynomial ring tensor a rank
$n$ exterior algebra. Using an extension of the Vandermonde determinant to
$\Omega_n$, the authors previously defined a family of doubly graded quotients
$\mathbb{W}_{n,k}$ of $\Omega_n$ which carry an action of the symmetric group
$\mathfrak{S}_n$ and satisfy a bigraded version of Poincar\'e Duality. In this
paper, we examine the duality modules $\mathbb{W}_{n,k}$ in greater detail. We
describe a monomial basis of $\mathbb{W}_{n,k}$ and give combinatorial formulas
for its bigraded Hilbert and Frobenius series. These formulas involve new
combinatorial objects called {\em ordered superpartitions}. These are ordered
set partitions $(B_1 \mid \cdots \mid B_k)$ of $\{1,\dots,n\}$ in which the
non-minimal elements of any block $B_i$ may be barred or unbarred.
