  Metrically homogeneous graphs are connected graphs which, when endowed with
the path metric, are homogeneous as metric spaces. In this paper we introduce
the concept of twisted automorphisms, a notion of isomorphism up to a
permutation of the language. We find all permutations of the language which are
associated with twisted automorphisms of metrically homogeneous graphs. For
each non-trivial permutation of this type we also characterize the class of
metrically homogeneous graphs which allow a twisted isomorphism associated with
that permutation. The permutations we find are, remarkably, precisely those
found by Bannai and Bannai in an analogous result in the context of finite
association schemes.
