Every Possible Cycle Length

Pyth, 11 8 bytes

.<W-0zz1

A lot more boring than my previous answer.

Every number that contains a 0 digit maps to itself. Any other number maps to the number that has its digits rotated by 1. So for example:

1 -> 1
10 -> 10
15 -> 51 -> 15
104 -> 104
123 -> 231 -> 312 -> 123

Python 2, 56 55 54 bytes

n=input()
a=b=1
while a+b<=n:a+=b;b+=1
print(n+~a)%b+a

Here's the first 21 outputs:

[1, 3, 2, 6, 4, 5, 10, 7, 8, 9, 15, 11, 12, 13, 14, 21, 16, 17, 18, 19, 20]

The pattern is obvious if we break the list up into chunks like so:

 1    2  3    4  5  6    7  8  9  10    11  12  13  14  15    16  17  18  19  20  21
[1]  [3, 2]  [6, 4, 5]  [10, 7, 8, 9]  [15, 11, 12, 13, 14]  [21, 16, 17, 18, 19, 20]

Pyth, 25 bytes

+hK/*J/h@h*8tQ2 2tJ2%-QKJ

This is the same sequence as @Sp3000, but with a closed form. The closed form is: