  An orientation-preserving branched covering $f: S^2 \to S^2$ is a nearly
Euclidean Thurston (NET) map if each critical point is simple and its
postcritical set has exactly four points. Inspired by classical, non-dynamical
notions such as Hurwitz equivalence of branched covers of surfaces, we develop
invariants for such maps. We then apply these notions to the classification and
enumeration of NET maps. As an application, we obtain a complete classification
of the dynamic critical orbit portraits of NET maps.
