77hedron

This 4-regular polyhedron has a curious combo of properties. Each face has 3, 4 or 5 sides, and two adjacent faces never have the same number of sides. Each vertex is incident with exactly two pentagonal faces and at least one triangular face. It has an Eulerian tour such that no two consecutive edges in the tour are incident with the same face. I.e., if you move along the edges and "move straight through" at each node, you will visit every edge before returning to where you started. On top of this, it has the rotational symmetries of a regular triangle.