  A graph $X$ is said to be "unstable" if the direct product $X \times K_2$
(also called the canonical double cover of $X$) has automorphisms that do not
come from automorphisms of its factors $X$ and $K_2$. It is "nontrivially
unstable" if it is unstable, connected, and nonbipartite, and no two distinct
vertices of X have exactly the same neighbors.
  We find three new conditions that each imply a circulant graph is unstable.
(These yield infinite families of nontrivially unstable circulant graphs that
were not previously known.) We also find all of the nontrivially unstable
circulant graphs of order $2p$, where $p$ is any prime number.
  Our results imply that there does not exist a nontrivially unstable circulant
graph of order $n$ if and only if either $n$ is odd, or $n < 8$, or $n = 2p$,
for some prime number $p$ that is congruent to $3$ modulo $4$.
