I am seeking a listing of the distinct Hamiltonian cycles following the edges of the icosahedron and the dodecahedron. By distinct I mean they are not congruent by some symmetry of the icosahedron or dodecahedron (respectively). So they do not make the same sequence of angular turns. For example (as Gerhard corrected me in the comments), there is just one distinct Hamiltonian cycle on the cube.
Hamiltonian cycles of the Platonic solids are all over the web, but I am not finding a definitive list of the number and a description of each. Thanks to anyone who can point me in the right direction!