  We study the orbits of $G=\mathrm{GL}(V)$ in the enhanced nilpotent cone
$V\times\mathcal{N}$, where $\mathcal{N}$ is the variety of nilpotent
endomorphisms of $V$. These orbits are parametrized by bipartitions of $n=\dim
V$, and we prove that the closure ordering corresponds to a natural partial
order on bipartitions. Moreover, we prove that the local intersection
cohomology of the orbit closures is given by certain bipartition analogues of
Kostka polynomials, defined by Shoji. Finally, we make a connection with Kato's
exotic nilpotent cone in type C, proving that the closure ordering is the same,
and conjecturing that the intersection cohomology is the same but with degrees
doubled.
