  We develop a new framework for analysing finite connected, oriented graphs of
valency 4, which admit a vertex-transitive and edge-transitive group of
automorphisms preserving the edge orientation. We identify a sub-family of
"basic" graphs such that each graph of this type is a normal cover of at least
one basic graph. The basic graphs either admit an edge-transitive group of
automorphisms that is quasiprimitive or biquasiprimitive on vertices, or admit
an (oriented or unoriented) cycle as a normal quotient. We anticipate that each
of these additional properties will facilitate effective further analysis, and
we demonstrate that this is so for the quasiprimitive basic graphs. Here we
obtain strong restirictions on the group involved, and construct several
infinite families of such graphs which, to our knowledge, are different from
any recorded in the literature so far. Several open problems are posed in the
paper.
