Today in lecture we learned that the transpose of a permutation matrix is the inverse of the permutation matrix. Meaning,
$$P^{T}P = I$$

I can work out the math by matrix multiplication but I'd prefer a deeper, more intuitive understanding.

What I have so far in my head is:

We know that the matrix $P$ will swap rows when we apply it to a matrix, let's say $A$. Then $PA$ will swap the $i^{th}$ row of A with the $j^{th}$ of $A$.

This then means that $P^{T}(PA)$ must swap our new $i^{th}$ row with the new $j^{th}$ row so we can have our original $A$ matrix back. Why is this always true? More specifically why does $P^{T}$ swap back out rows...?