data = {{0, 1}, {5 Pi/6, 1}, {8 Pi/3, 1}, {7 Pi/2, 1}, {10 Pi/3,
    1}, {25 Pi/6, 1}, {6 Pi, 1}, {41 Pi/6, 1}, {20 Pi/3, 1}, {15 Pi/2,
     1}, {28 Pi/3, 1}, {61 Pi/6, 1}};

ListPolarPlot[data]

label[data_, acc_: 0.0001] := Module[{c},
  c[_] = 0;
  ++c[{
       Round[Mod[First[N@#], 2 Pi], acc],
       Round[Last[N@#], acc]
       }] & /@ data
  ]

labels = label[data];
groups = Map[Last, GatherBy[Transpose[{labels, data}], First], {2}];

ListPolarPlot[groups, PlotMarkers -> {"OpenMarkers", Large}]