A069889 Self-inverse permutation of natural numbers induced by the automorphism RotateHandshakes_et_DeepReverse! acting on the parenthesizations encoded by A014486.

0, 1, 3, 2, 7, 8, 6, 4, 5, 17, 20, 18, 21, 22, 16, 19, 14, 9, 11, 15, 10, 12, 13, 45, 54, 48, 57, 61, 46, 55, 49, 58, 62, 50, 59, 63, 64, 44, 53, 47, 56, 60, 42, 51, 37, 23, 28, 39, 25, 30, 33, 43, 52, 38, 24, 29, 40, 26, 31, 34, 41, 27, 32, 35, 36, 129, 157, 138, 166, 180