  We consider the properties of the Moore-Read Pfaffian state under
particle-hole conjugation. We show that the particle-hole conjugate of the
Pfaffian state - or "anti-Pfaffian" state - is in a different universality
class from the Pfaffian state, with different topological order. The two states
can be distinguished by both their bulk and edge physics though the difference
is most dramatic at the edge: the edge of the anti-Pfaffian state has a
composite structure that leads to a different thermal Hall conductance and
different tunneling exponents than the Pfaffian state. At the same time, the
two states are exactly degenerate in energy for a $\nu = 5/2$ quantum Hall
system in the idealized limit of zero Landau level mixing. Thus, both are good
candidates for the observed $\nu = 5/2$ quantum Hall plateau.
