It is a distance-transitive graph (see the Foster census) and therefore distance regular.

By subdividing the cycle edges into two matchings, we can partition the Heawood graph into three perfect matchings (that is, 3-color its edges) in eight different ways.

[2] Every two perfect matchings, and every two Hamiltonian cycles, can be transformed into each other by a symmetry of the graph.

The result is the regular map {6,3}2,1, with 7 hexagonal faces.

With this interpretation, the 6-cycles in the Heawood graph correspond to triangles in the Fano plane.

Also, the Heawood graph is the Tits building of the group SL3(F2).

The Heawood graph has crossing number 3, and is the smallest cubic graph with that crossing number (sequence A110507 in the OEIS).

[11] It acts transitively on the vertices, on the edges and on the arcs of the graph.