  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 be 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.
