  Let $\vec C_m$ and $\vec C_n$ be directed cycles of length $m$ and $n$, with
$m,n \ge 3$, and let $P(\vec C_m \mathbin{\Box} \vec C_n)$ be the digraph that
is obtained from the Cartesian product $\vec C_m \mathbin{\Box} \vec C_n$ by
choosing a vertex $v$, and reversing the orientation of all four directed edges
that are incident with $v$. (This operation is called "pushing" at the vertex
$v$.) By applying a special case of unpublished work of S.X.Wu, we find
elementary number-theoretic necessary and sufficient conditions for the
existence of a hamiltonian cycle in $P(\vec C_m \mathbin{\Box} \vec C_n)$.
  A consequence is that if $P(\vec C_m \mathbin{\Box} \vec C_n)$ is
hamiltonian, then $\gcd(m,n) = 1$, which implies that $\vec C_m \mathbin{\Box}
\vec C_n$ is not hamiltonian. This final conclusion verifies a conjecture of
J.B.Klerlein and E.C.Carr.
