  A polynomial is called self-reciprocal (or palindromic) if the sequence of
its coefficients is palindromic. In this paper we enumerate self-reciprocal
irreducible monic polynomials over a finite field with prescribed leading
coefficients. Asymptotic expression with explicit error bound is derived, which
is used to show that such polynomials with degree $2n$ always exist provided
that the number of prescribed leading coefficients is slightly less than $n/4$.
Exact expressions are also obtained for fields with two or three elements and
up to two prescribed leading coefficients.
