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If you look at the documentation of the relabel method (by typing U.relabel?), you can see that there is a return_map option`:

sage: U.relabel(return_map=True)                                                                                                                                                                             
{(): 0,
 (1,5,4,3,2): 1,
 (1,4,5,3,2): 2,
 (1,3,2): 3,
 (1,2,5,4,3): 4,
 (2,4,5): 5,
 (1,5)(2,3): 6,
 (1,4,3,2,5): 7,
 (1,3,2,4,5): 8,
 (1,2,3): 9,
 (2,3,5): 10,
 (1,5)(3,4): 11,
 (1,4,5): 12,
 (1,3,5): 13,
 (1,2)(3,4): 14,
 (2,5,4): 15,
 (1,5,3,2,4): 16,
 (1,4)(2,3): 17,
 (1,3,2,5,4): 18,
 (1,2,4,5,3): 19,
 (3,5,4): 20,
 (1,5,3,4,2): 21,
 (1,4,2): 22,
 (1,3,5,4,2): 23,
 (1,2,5,3,4): 24,
 (2,4,3): 25,
 (1,5,4,2,3): 26,
 (1,4,2,5,3): 27,
 (1,3)(2,4): 28,
 (1,2,3,5,4): 29,
 (2,3)(4,5): 30,
 (1,5,3): 31,
 (1,4,3): 32,
 (1,3)(4,5): 33,
 (1,2)(3,5): 34,
 (2,5)(3,4): 35,
 (1,5)(2,4): 36,
 (1,4,2,3,5): 37,
 (1,3,4,2,5): 38,
 (1,2,4): 39,
 (3,4,5): 40,
 (1,5,2): 41,
 (1,4,3,5,2): 42,
 (1,3,4,5,2): 43,
 (1,2,5): 44,
 (2,4)(3,5): 45,
 (1,5,2,3,4): 46,
 (1,4)(2,5): 47,
 (1,3,5,2,4): 48,
 (1,2,3,4,5): 49,
 (2,3,4): 50,
 (1,5,4): 51,
 (1,4)(3,5): 52,
 (1,3,4): 53,
 (1,2)(4,5): 54,
 (2,5,3): 55,
 (1,5,2,4,3): 56,
 (1,4,5,2,3): 57,
 (1,3)(2,5): 58,
 (1,2,4,3,5): 59}