Galois groups of reciprocal polynomials II: Twisted reciprocal polynomials

We study the Galois group $G_f$ of a random polynomial $f$ of height at most $H$ in the family of polynomials of degree $2n$ satisfying the twisted reciprocal relation $f(x)=x^{2n}\mathrm{/}{b}^n\cdot f(b\mathrm{/}x)$, which arise in a wide variety of applications. Our main result is a theorem of van der Waerden-Bhargava type: the probability that $G_f$ is not the full hyperoctahedral group $S_2\wr S_n$ is $\mathrm{\Theta }(H^{-1}\log H)$, independent of $b$, with the leading-order group $G_1$ being of index $2$. This paper is a companion to a recent paper by the authors and Bertelli addressing reciprocal polynomials (i.e. the case $b=1$).