  We study finite four-valent graphs Gamma admitting an edge-transitive group G
of automorphisms such that G determines and preserves an edge-orientation on
Gamma, and such that at least one G-normal quotient is a cycle (a quotient
modulo the orbits of a normal subgroup of G). We show on the one hand that the
number of distinct cyclic G-normal quotients can be unboundedly large. On the
other hand existence of independent cyclic G-normal quotients (that is, they
are not extendable to a common cyclic G-normal quotient) places severe
restrictions on the graph Gamma and we classify all examples. We show there are
five infinite families of such pairs (Gamma, G), and in particular that all
such graphs have at least one normal quotient which is an unoriented cycle. We
compare this new approach with existing treatments for the sub-class of weak
metacirculant graphs with these properties, finding that only two infinite
families of examples occur in common from both analyses. Several open problems
are posed.
