  A Littlewood polynomial is a single-variable polynomial all of whose
coefficients lie in $\{ \pm 1\}$. We establish the leading term asymptotics of
the number of reciprocal or skew-reciprocal Littlewood polynomials with square
discriminant. This relates to a bounded-height analogue of the Van der Waerden
conjecture on Galois groups of random polynomials.
