Let $A\in\mathbb{R}^{n\times n}$ be any matrix and let $P\in\mathbb{R}^{n\times n}$ be a permutation matrix.
Left multiplication of $A$ by $P$ corresponds to permuting the rows of $P$. On the other hand, right multiplication of $A$ by $P$ corresponds to permuting the columns of $P$. However, I'm not sure how these operations affect the eigenvalues and eigenvectors.
In other words, how are the eigenvalues and eigenvectors of $A$, $PA$, and $AP$ related or not?
Well, we first note that the eigenvalues do change, just consider a quick example: $$ A=\begin{pmatrix}-1 &0\\0&1\end{pmatrix} \Rightarrow \lambda_{1,2}=\pm1, \text{ while } PA=\begin{pmatrix}0&1\\-1&0\end{pmatrix} \Rightarrow \lambda_{1,2}=\pm i$$ However, the product of all the eingenvalues (counting multiplicity) will only change its signal when applying a permutation (or will not change at all if you have a even permutation). That's true because the determinant of a matrix is equal to the product of its eigenvalues.