Curver solves the conjugacy problem for periodic mapping classes by computing their quotient orbifold (and covering map), a total conjugacy invariant. You can perform this calculation here, where the covering map is described by the rotation numbers about the cone points.
To do this, choose a surface Sg,n by specifying a genus and number of punctures (we require that g ≥ 0, n > 0 and 6g - 3n + 6 > 0). Then pick some words (separated by ;) corresponding to a periodic mapping class in the generating set:
Computed using Curver 0.4.1.