We investigate the subclass of reversible functions that are self-inverse and relate them to reversible circuits that are equal to their reverse circuit, which are called palindromic circuits. We precisely determine which self-inverse functions can b
e realized as a palindromic circuit. For those functions that cannot be realized as a palindromic circuit, we find alternative palindromic representations that require an extra circuit line or quantum gates in their construction. Our analyses make use of involutions in the symmetric group $S_{2^n}$ which are isomorphic to self-inverse reversible function on $n$ variables.
