  In this paper, we begin by partitioning the edges (or arcs) of a circulant
(di)graph according to which generator in the connection set leads to each
edge. We then further refine the partition by subdividing any part that
corresponds to an element of order less than $n$, according to which of the
cycles generated by that element the edge is in. It is known that if the
(di)graph is connected and has no multiple edges, then any automorphism that
respects the first partition and fixes the vertex corresponding to the group
identity must be an automorphism of the group (this is in fact true in the more
general context of Cayley graphs). We show that automorphisms that respect the
second partition and fix 0 must also respect the first partition, so are again
precisely the group automorphisms of $\mathbb Z_n$.
