A 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. We show that up to equivalence, each NET map admits a normal form in terms of simple affine data. This data can then be used as input for algorithms developed for the computation of fundamental invariants, now systematically tabulated in a large census.