You are asking about the signature of the permutation $\pi_{m,n}: \mathbb Z/n\mathbb Z\to \mathbb Z/n\mathbb Z :x\mapsto [m]x$. The answer for $n$ odd is $sgn (\pi_{m,n}) =(\frac{m}{n}) $ where $ (\frac{m}{n})$ is the Jacobi symbol.
In case $n$ is even you get $sgn (\pi_{m,n})=(-1)^{(\frac{n}{2}+1)(\frac{m-1}{2})} $
Here is the plan for proving this.
Step 1 Assume $n$ is a prime $p$. Then the result $sgn (\pi_{m,p}) =(\frac{m}{p})$, where $(\frac{m}{p})$ is just the Legendre symbol, is due to Zolotarev and you can find a proof in the link provided by Bill in his comment to the question. [The linked article states that it is easy to generalize from prime $n$ to odd $n$, but says nothing about even $n$]
Step 2 Prove the general case by noticing that the function $sgn (\pi_{m,n})$ of $m$ and $n$ satisfies exactly the same multiplicative identities as the Jacobi symbol. Details are in this document (in French)