I feel like the problem should have been studied, but I wasn't able to find anything precise.
Given two bracelets with $n$ beads and $m$ colors, given that the multiplicity of each color is the same, and excluding the obvious rotation to $n$ positions plus a reflection (for a total of $2n$ trials with $n$ comparisons, so with a complexity of $\mathcal{O}(n^2)$), what are better known algorithms to decide whether the two bracelets are isomorphic?
For now, the only alternative that came to mind was making a polar Discrete Fourier Transform and then computing the ratio of the two to get the rotation angle before comparing the two bracelets.