In this paper we study finite, connected, 4-valent graphs X which admit an action of a group G which is transitive on vertices and edges, but not transitive on the arcs of X. Such a graph X is said to be (G, 1/2)-transitive. The group G induces an orientation of the edges of X, and a certain class of cycles of X (called alternating cycles) determined by the group G is identified as having an important influence on the structure of X. The alternating cycles are those in which consecutive edges have opposite orientations. It is shown that X is a cover of a finite, connected, r-valent, (G, 1/2)-transitive graph for which the alternating cycles have one of three additional special properties, namely they are tightly attached, loosely attached, or antipodally attached. We give examples with each of these special attachment properties and moreover we complete the classification (begun in a separate paper by the first author) of the tightly attached examples. (C) 1999 Academic Press.