Take a bracelet with colored beads on it. Normally two bracelets belong to the same equivalence class under rotations and reflections. For an example, consider the bracelet denoted by the word abcdec:
cabcde ecabcd decabc cdecab ... (rotations)
cedcba dcbace ... (reflections)
Consider the larger class where two bracelets are equivalent if all local bead pairings are preserved. From our example we have the multiset {ab, bc, cd, de, ec, ca}. The right-left ordering, ba vs ab doesn't matter. Clearly rotations and reflections preserve bead pairings, but so does:
abcedc
where we've transposed the e and the d. How do you describe this symmetry?
We can interpret the bead pairings as the edges in a multi-graph whose vertices are the bead labels. Each words in the corresponding equivalence class corresponds to the sequence of vertices in a Euler circuit of the graph, and vice versa.
We get a one-to-one correspondence if we consider multiple edges between a pair of vertices to be indistinguishable, so that a Euler circuit is completely determined by its vertices.