  We consider pseudo-involutions in the Riordan group where the generating
function $g$ for the first column of a Riordan array satisfies a palindromic or
near-palindromic functional equation. For those types of equations, we find,
for very little work, the pseudo-involutory companion of $g$ and have a
pseudo-involution in a $k$-Bell subgroup. There are only slight differences in
the ordinary and exponential cases. In many cases, we also develop a general
method for finding B-functions of Riordan pseudo-involutions in $k$-Bell
subgroups, and show that these B-functions involve Chebyshev polynomials. We
apply our method for many families of Riordan arrays, both new and already
known.
  We also have some duality and reciprocity results. Since many of the examples
we discuss have combinatorial significance, we conclude with a few remarks on
the general framework for a combinatorial interpretation of some of the
generating function results we obtain.
