What is spectrum permutation?
Reminder what a permutation is: The exchange of indices in a sequence. 

That's exactly what they're doing here.

Invertible in this context (only read the paragraph so far) means 

"the multiplication with $\sigma$ under mod $n$ is invertible, i.e. given the result of that multiplication and $\sigma$, you can caclulate the original value in all cases." 

Let's do an example:

let's say we're  $x\in \mathbb Z 
\setminus 13$, and you multiply by $\sigma = 7$, you get $x\sigma = 4 \mod 13$. You know that $x=8$, undisputably! (because 7·8  = 56, and 56 mod 13 = 4, and it's only that easy because this is a prime field).

That works for any possible result $xz\in \mathbb Z\setminus 13$, because $\sigma$ is coprime to 13; so, $\sigma$ is invertible for $(\cdot,\mathbb Z \setminus 13)$.

I picked that 13 because it's prime, and prime numbers are by definition coprime to any smaller integer. 

Let's do an counter-example of that:

For FFTs, the length is usually not a prime number, but can be any natural integer. $n=2^m, m \in \mathbb N$ is a very common FFT length (because you can decompose it easily using simple butterflies); then, not every $n$ is invertible in the sense above: for example, if your multiplication $x\cdot4 = 0 \mod 32$, you can't say whether $x=0$, $x=8$ or $x=16$; it's not invertible.