  It is well-known that every 3-connected planar graph has a unique planar
embedding on the sphere. We study the extension to triangulated 1-planar
graphs, T1P graphs for short, which admit an embedding in which each edge is
crossed at most once and each face is a triangle, and obtain an algorithmic
solution by a cubic time recognition algorithm that also counts the number of
T1P embeddings. In particular, we show that every triangulated planar graph has
a unique T1P embedding, although it may admit many 1-planar embeddings, and
that any 6-connected T1P graph has a unique 1-planar embedding, except for full
generalized two-stars that admit two or eight 1-planar embeddings. Our
algorithm extends, refines, and corrects a previous recognition algorithm by
Chen, Grigni and Papadimitiou (``Recognizing Hole-Free 4-Map Graphs in Cubic
Time'', Algorithmica 45 (2006)).
