  We analyze self-dual polyhedral cones and prove several properties about
their slack matrices. In particular, we show that self-duality is equivalent to
the existence of a positive semidefinite (PSD) slack. Beyond that, we show that
if the underlying cone is irreducible, then the corresponding PSD slacks are
not only doubly nonnegative matrices (DNN) but are extreme rays of the cone of
DNN matrices, which correspond to a family of extreme rays not previously
described. More surprisingly, we show that, unless the cone is simplicial, PSD
slacks not only fail to be completely positive matrices but they also lie
outside the cone of completely positive semidefinite matrices. Finally, we show
how one can use semidefinite programming to probe the existence of self-dual
cones with given combinatorics. Our results are given for polyhedral cones but
we also discuss some consequences for negatively self-polar polytopes.
