The family $\mathcal{OG}(4)$ consisting of graph-group pairs $(Γ, G)$, where $Γ$ 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 $(Γ, G)$ 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).