  We consider an interesting class of combinatorial symmetries of polytopes
which we call \emph{edge-length preserving combinatorial symmetries}. These
symmetries not only preserve the combinatorial structure of a polytope but also
map each edge of the polytope to an edge of the same length. We prove a simple
sufficient condition for a polytope to realize all edge-length preserving
combinatorial symmetries by isometries of ambient space. The proof of this
condition uses Cauchy's rigidity theorem in an unusual way.
