  Let $\mathcal{OG}(4)$ denote the family of all graph-group pairs $(\Gamma,G)$
where $\Gamma$ is 4-valent, connected and $G$-oriented
($G$-half-arc-transitive). Using a novel application of the structure theorem
for biquasiprimitive permutation groups of the second author, we produce a
description of all pairs $(\Gamma, G) \in\mathcal{OG}(4)$ for which every
nontrivial normal subgroup of $G$ has at most two orbits on the vertices of
$\Gamma$. In particular we show that $G$ has a unique minimal normal subgroup
$N$ and that $N \cong T^k$ for a simple group $T$ and $k\in \{1,2,4,8\}$. This
provides a crucial step towards a general description of the long-studied
family $\mathcal{OG}(4)$ in terms of a normal quotient reduction. We also give
several methods for constructing pairs $(\Gamma, G)$ of this type and provide
many new infinite families of examples, covering each of the possible
structures of the normal subgroup $N$.
