  We show that a cone admits a gauge-reversing map if and only if it is a
symmetric cone. We use this to prove that every isometry of a Hilbert geometry
is a collineation unless the Hilbert geometry is the projective space of a
non-Lorentzian symmetric cone, in which case the collineation group is of index
two in the isometry group. We also determine the isometry group of the Thompson
geometry on a cone.
