Is it possible to write the same permutation as a collection of disjoint cycles in two different ways? The answer is apparently no. I just don't get it. 
$(1, 2, 3)(4, 5)$ and $(2, 3, 1)(5, 4)$ are the same permutation. Aren't those $2$ different ways of writing the same permutation? 
 A: The misconception is that $(1\ 2\ 3)$ is different from $(2\ 3\ 1)$. Although the markup looks different, they represent the exact same cycle. In the same manner we have $(4\ 5) = (5\ 4)$.
To get a somewhat unique representation of the cycles, you have to impose a condition on the markup, for example to start with the smallest element (wich is not mapped to itself). Then the canonical representation would in fact be your first one, since $1$ is the smallest element of $(1\ 2\ 3)$ and $4$ is the smallest element of $(4\ 5)$.
A: You are right that they are the same permutation. This is not a contradiction; you simply wrote the same cycles in two different ways. $(231)=(123)$ and $(54)=(45)$. This is because $231$ is a cyclic permutation of 123 and 45 is a cyclic permutation of 54. You get 231 from 123 by shifting everything to the left by one place (then the 1 ends up at the end).
"Rearrangement" here refers to rearranging the order of the cycles themselves, not the elements within the cycles.
