Seems to be really addictive, so you will have to endure one more answer I am afraid.

What follows is actually present in several of already given answers, I am just trying to make it as simple as possible.

Write down $\binom n2$ statements "$1<2$", "$1<3$", ..., "$n-1<n$".

Now perform a permutation and count how many of these statements will become violated.

If this permutation is a transposition  $(ij)$, then those violated are all "$i<k$" with $k<j$, all "$\ell<j$" with $\ell>i$ (same number twice), and $i<j$. So each transposition violates an odd number of these statements.

Viewing result of a permutation as a reordering, we see that performing one more transposition switches between "even violators" and "odd violators".