  Finite quasi semimetrics on $n$ can be thought of as nonnegative valuations
on the edges of a complete directed graph on $n$ vertices satisfying all
possible triangle inequalities. They comprise a polyhedral cone whose symmetry
groups were studied for small $n$ by Deza, Dutour and Panteleeva. We show that
the symmetry and combinatorial symmetry groups are as they conjectured.
Integral quasi semimetrics have apecial place in the theory of tiled orders,
being known as exponent matrices, and can be viewed as monoids under
componentwise maximum; we provide a novel derivation of the automorphism group
of that monoid. Some of these results follow from more general consideration of
polyhedral cones that are closed under componentwise maximum.
