Suppose that $\mathfrak{g}$ is an $n$-dimensional complex semisimple Lie algebra, with a root system $R$, and suppose that $S = \{\alpha_1,...,\alpha_m\}$ is a base for $R$.
The Cartan matrix for $R$ is the $m$-dimensional matrix with entries $c_{ij} = \frac{2(\alpha_i,\alpha_j)}{(\alpha_i,\alpha_i)}$.
Reordering of the elements of the chosen base clearly changes the entries of the matrix. Specifically, suppose that $\bar{S} = \{\alpha_{\sigma (1)},...,\alpha_{\sigma(m)}\}$ for $\sigma \in S_m$. Now we have a different Cartan matrix, with entries $\bar{c}_{ij} = \frac{2(\alpha_{\sigma(i)},\alpha_{\sigma(j)})}{(\alpha_{\sigma(i)},\alpha_{\sigma(i)})}$.
What does this specifically do to the Cartan matrix? I think that it permutes the diagonals, and playing with some examples seems to confirm that (as I think that it should), but I can't get my head around why.