We introduce the palindromic automorphism group and the palindromic Torelli group of a right-angled Artin group $A\_\Gamma$. The palindromic automorphism group $\Pi A\_\Gamma$ is related to the principal congruence subgroups of GL$(n,\mathbb Z)$ and to the hyperelliptic mapping class group of an oriented surface, and sits inside the centraliser of a certain hyperelliptic involution in Aut$(A\_\Gamma)$. We obtain finite generating sets for $\Pi A\_\Gamma$ and for this centraliser, and determine precisely when these two groups coincide. We also find generators for the palindromic Torelli group.