What does the term “nontrivial rotation” mean? The paper “Gimli: a cross-platform permutation” contains the following information:  


  Occasionally (after rounds 24, 20, 16, etc.) Gimli adds an asymmetric constant to entry 0 of the first row. This constant has many bits set (it is essentially the golden ratio 0x9e3779b9, as used in TEA), and is not close to any of its nontrivial rotations (never fewer than 12 bits different)  


What does the term “nontrivial rotation” mean? How to distinguish a “nontrivial” rotation from a “trivial” one?
A trivial rotation is one where the input to the rotation is equal to the output (hence the rotation is equal to the identity function). Rotating by a multiple of $n$ is a trivial rotation for all $n$ bit values. Note that for certain values other trivial rotations may exist. See below for the values of the (right) rotations of 0x9e3779b9 ($n = 32$) and the amount of different bits for each:

0:  0x9e3779b9 (0  bits different)
1:  0xcf1bbcdc (14 bits different)
2:  0x678dde6e (22 bits different)
3:  0x33c6ef37 (18 bits different)
4:  0x99e3779b (12 bits different)
5:  0xccf1bbcd (14 bits different)
6:  0xe678dde6 (18 bits different)
7:  0x733c6ef3 (16 bits different)
8:  0xb99e3779 (14 bits different)
9:  0xdccf1bbc (12 bits different)
10: 0x6e678dde (16 bits different)
11: 0x3733c6ef (16 bits different)
12: 0x9b99e377 (16 bits different)
13: 0xcdccf1bb (14 bits different)
14: 0xe6e678dd (12 bits different)
15: 0xf3733c6e (16 bits different)
16: 0x79b99e37 (20 bits different)
17: 0xbcdccf1b (16 bits different)
18: 0xde6e678d (12 bits different)
19: 0xef3733c6 (14 bits different)
20: 0x779b99e3 (16 bits different)
21: 0xbbcdccf1 (16 bits different)
22: 0xdde6e678 (16 bits different)
23: 0x6ef3733c (12 bits different)
24: 0x3779b99e (14 bits different)
25: 0x1bbcdccf (16 bits different)
26: 0x8dde6e67 (18 bits different)
27: 0xc6ef3733 (14 bits different)
28: 0xe3779b99 (12 bits different)
29: 0xf1bbcdcc (18 bits different)
30: 0x78dde6e6 (22 bits different)
31: 0x3c6ef373 (14 bits different)