The odd antiprisms are both Eulerian and polyhedral, with the first implying that the edge can be represented with a single closed path. The Cuboctahedron also has that property. With the rule to always go straight through a crossing, the cuboctahedron splits into 4 loops. The odd anti-prisms each use just one loop.
The quartic 9-5 graph also uses just one loop -- the eulerian cycle can be found by always going straight through each intersection.
Are there any triangulated graphs with this property?
Yes.
This planar triangulation has an Eulerian cycle formed by going straight across each vertex:
There are 13 vertices; 7 are 4-valent, 5 are 6-valent, and 1 is 8-valent. There are 33 edges and 22 faces.
I find it easier to work with the dual, where we need to find a circuit crossing all the edges, going straight across each face.
Here it is:
You might appreciate the graph6 code of the triangulation:
or its dual:
These can be imported into Mathematica, etc (
PlanarGraph[ImportString["…", "graph6"]].)