I still have problems with the Union statement and it may not always remove duplicates on Lists. So, I combined the last 2 methods and it gives me the very readable code (by the way, this is an application to crystallography when we fold the atomic positions into one unit cell):
data = {{0.416668, 0.295182, 0.044696}, {0.416668, 1.29518,
0.044696}, {1.41667, 1.29518, 0.044696}, {0.295182, 0.044696,
0.416668}, {0.295182, 1.0447, 0.416668}, {1.29518, 1.0447,
0.416668}, {0.044696, 0.416668, 0.295182}, {0.044696, 1.41667,
0.295182}, {1.0447, 1.41667,
0.295182}, {-0.416668, -0.295182, -0.044696}, {-0.416668,
0.704818, -0.044696}, {0.583332,
0.704818, -0.044696}, {-0.295182, -0.044696, -0.416668}, \
{-0.295182, 0.955304, -0.416668}, {0.704818,
0.955304, -0.416668}, {-0.044696, -0.416668, -0.295182}, \
{-0.044696, 0.583332, -0.295182}, {0.955304, 0.583332, -0.295182}}
ClearAll[FoldNegative, FoldPositive]
FoldNegative[x_] := If[x < 0, x + 1, x]
FoldPositive[x_] := If[x >= 1, x - 1, x]
SetAttributes[FoldNegative, Listable]
SetAttributes[FoldPositive, Listable]
Print["Corrected Data"]
data3 = FoldPositive[FoldNegative[data]]
Print["Final Data"]
DeleteDuplicates[data3, SameQ[Round[#1, 10^-4], Round[#2, 10^-4]] &]
The Output is
{{0.416668, 0.295182, 0.044696}, {0.416668, 1.29518,
0.044696}, {1.41667, 1.29518, 0.044696}, {0.295182, 0.044696,
0.416668}, {0.295182, 1.0447, 0.416668}, {1.29518, 1.0447,
0.416668}, {0.044696, 0.416668, 0.295182}, {0.044696, 1.41667,
0.295182}, {1.0447, 1.41667,
0.295182}, {-0.416668, -0.295182, -0.044696}, {-0.416668,
0.704818, -0.044696}, {0.583332,
0.704818, -0.044696}, {-0.295182, -0.044696, -0.416668}, {-0.295182,
0.955304, -0.416668}, {0.704818,
0.955304, -0.416668}, {-0.044696, -0.416668, -0.295182}, {-0.044696,
0.583332, -0.295182}, {0.955304, 0.583332, -0.295182}}
Corrected Data
{{0.416668, 0.295182, 0.044696}, {0.416668, 0.29518,
0.044696}, {0.41667, 0.29518, 0.044696}, {0.295182, 0.044696,
0.416668}, {0.295182, 0.0447, 0.416668}, {0.29518, 0.0447,
0.416668}, {0.044696, 0.416668, 0.295182}, {0.044696, 0.41667,
0.295182}, {0.0447, 0.41667, 0.295182}, {0.583332, 0.704818,
0.955304}, {0.583332, 0.704818, 0.955304}, {0.583332, 0.704818,
0.955304}, {0.704818, 0.955304, 0.583332}, {0.704818, 0.955304,
0.583332}, {0.704818, 0.955304, 0.583332}, {0.955304, 0.583332,
0.704818}, {0.955304, 0.583332, 0.704818}, {0.955304, 0.583332,
0.704818}}
Final Data
{{0.416668, 0.295182, 0.044696}, {0.295182, 0.044696,
0.416668}, {0.044696, 0.416668, 0.295182}, {0.583332, 0.704818,
0.955304}, {0.704818, 0.955304, 0.583332}, {0.955304, 0.583332,
0.704818}}