  The family $\mathcal{OG}(4)$ consisting of graph-group pairs $(\Gamma, G)$,
where $\Gamma$ is a finite, connected, 4-valent graph admitting a $G$-vertex-,
and $G$-edge-transitive, but not $G$-arc-transitive action, has recently been
examined using a normal quotient methodology. A subfamily of $\mathcal{OG}(4)$
has been identified as `basic', due to the fact that all members of
$\mathcal{OG}(4)$ are normal covers of at least one basic pair. We provide an
explicit classification of those basic pairs $(\Gamma, G)$ which have at least
two independent cyclic $G$-normal quotients (these are $G$-normal quotients
which are not extendable to a common cyclic normal quotient).
