Consider an $n\times n$ square matrix A.
Let $B = A^{T}$
We know that $|B|$ is the sum of n-component products $b_{1i}b_{2j}....b_{nr}$ where $(i, j, ..., r)$ is a permutation of the natural order $(1, 2, ..., n)$, with a plus or minus sign attached to $b_{1i}b_{2j}....b_{nr}$ if $(i, j, ..., r)$ is an even permutation or an odd permutation, respectively.
Now,
\begin{equation}
b_{1i}b_{2j}....b_{nr} = a_{i1}a_{j2}....a_{rn}
\end{equation}
where $a_{i1}a_{j2}....a_{rn}$ is in $|A|$. Note that the second subscripts in the left hand side of the equation are in the same permutation as the first subscripts in the right hand side of the equation.
G.Hadley says that if the terms in $a_{i1}a_{j2}....a_{rn}$ are rearranged so that the first subscripts are in the natural order, then 'by symmetry', the second subscripts will now have the same number of inversions as the permutation $(i,,j, ..., r)$. I am not able to understand what symmetry is invoked here and how it can be argued that the numbers of inversions is the same.