  The operation of switching a graph $\Gamma$ with respect to a subset $X$ of
the vertex set interchanges edges and non-edges between $X$ and its complement,
leaving the rest of the graph unchanged. This is an equivalence relation on the
set of graphs on a given vertex set, so we can talk about the automorphism
group of a switching class of graphs.
  It might be thought that switching classes with many automorphisms would have
the property that all their graphs also have many automorphisms. However the
main theorem of this paper shows a different picture: with finitely many
exceptions, if a non-trivial switching class $\mathcal{S}$ has primitive
automorphism group, then it contains a graph whose automorphism group is
trivial. We also find all the exceptional switching classes; up to
complementation, there are just six.
