Generate permutations partitioned by rotational equivalence

Question

This seems like a very inefficient way of doing what I want. I generate all permutations of (for example) {1, 1, 2, 2}, then for each permutation I generate its rotations, select the first one in Sort order, and use that for GatherBy.

list = {1, 1, 2, 2};
display@GatherBy[Permutations@list, Sort[Table[RotateLeft[#, i], {i, 0, Length@# - 1}]][[1]]& ]
(* {{"1122", "1221", "2112", "2211"}, {"1212", "2121"}} *)

How can I do this more efficiently?

Answer 1

GroupOrbits[CyclicGroup[Length @ #], Permutations @ #, Permute] & @ list

{{{1, 1, 2, 2}, {1, 2, 2, 1}, {2, 1, 1, 2}, {2, 2, 1, 1}}, 
 {{1, 2, 1, 2}, {2, 1, 2, 1}}}

Answer 2

With[{p = Permutations[list]}, Extract[p, #] & /@
  ConnectedComponents[MapIndexed[UndirectedEdge,
    Lookup[PositionIndex[p], RotateLeft[p, {0, 1}]]]]]

{{{1, 1, 2, 2}, {1, 2, 2, 1}, {2, 1, 1, 2}, {2, 2, 1, 1}}, {{1, 2, 1, 2}, {2, 1, 2, 1}}}