Summary: A fundamental result in topological graph theory by H. Whitney [2-isomorphic graphs, Am. J. Math. 55, 245–254 (1933; Zbl 0006.37005 and JFM 59.1235.01)] states that a 3-connected graph has at most one planar embedding. C. Thomassen [J. Comb. Theory, Ser. B 48, 155–177 (1990; Zbl 0704.05011)] generalized this to large-edge-width-embeddings on higher surfaces. We establish several unique embedding results for 3-connected graphs on orientable surfaces which admit relatively large facial walks and representativity and hence generalize Thomassen’s uniqueness theorem on large-edge-width-embeddings.