We enumerate certain geometric equivalence classes of subgraphs induced by Hamiltonian paths and cycles in complete graphs. These classes are orbits under the action of certain direct products of dihedral and cyclic groups on sets of strings representing subgraphs. The orbits are enumerated using Burnside's lemma. The technique used also provides an alternative proof of the formulae found by Golomb and Welch, which give the number of distinct n-gons on fixed, regularly spaced vertices up to rotation and optionally reflection.