  We study the Galois groups $G_f$ of degree $2n$ reciprocal (a.k.a.
palindromic) polynomials $f$ of height at most $H$, finding that $G_f$ falls
short of the maximal possible group $S_2 \wr S_n$ for a proportion of all $f$
bounded above and below by constant multiples of $H^{-1} \log H$, whether or
not $f$ is required to be monic. This answers a 1998 question of Davis-Duke-Sun
and extends Bhargava's 2023 resolution of van der Waerden's 1936 conjecture on
the corresponding question for general polynomials. Unlike in that setting, the
dominant contribution comes not from reducible polynomials but from those $f$
for which $(-1)^n f(1) f(-1)$ is a square, causing $G_f$ to lie in an index-$2$
subgroup.
