  Whitney proved that 3-connected planar graphs can be embedded uniquely on the
sphere. In general, such a graph may also have embeddings on different
surfaces. Enami focused on cubic graphs and showed that a 3-connected cubic
planar graph can be re-embedded on a surface of non-negative Euler
characteristic if and only if the dual graph contains a specific subgraph.
Motivated by applications to triangulated surfaces, we investigate strong
re-embeddings and show that these are characterised by a subset of Enami's
subgraphs. Additionally, we provide criteria for when a graph does not possess
a strong re-embedding on the projective plane, the torus or the Klein bottle.
