  A formula for the number of monic irreducible self-reciprocal polynomials, of
a given degree over a finite field, was given by Carlitz in 1967. In 2011
Ahmadi showed that Carlitz's formula extends, essentially without change, to a
count of irreducible polynomials arising through an arbitrary quadratic
transformation. In the present paper we provide an explanation for this
extension, and a simpler proof of Ahmadi's result, by a reduction to the known
special case of self-reciprocal polynomials and a minor variation. We also
prove further results on polynomials arising through a quadratic
transformation, and through some special transformations of higher degree.
