Galois groups of reciprocal polynomials and the van der Waerden-Bhargava theorem

We study the Galois groups Gf of degree $2n$ reciprocal (a.k.a. palindromic) polynomials f of height at most H, finding that Gf falls short of the maximal possible group S₂ Sₙ 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 Gf to lie in an index-$2$ subgroup.