  Let $n \geq 3$ be an integer. In this paper, we study the average behavior of
the $2$-torsion in class groups of number fields cut out by integral binary
$n$-ic forms having any fixed odd leading coefficient. Specifically, we compute
upper bounds on the average size of the $2$-torsion in the class groups of such
number fields. Conditional on a uniformity estimate, we further prove that each
of these upper bounds is in fact an equality.
  Our theorems extend recent work of Bhargava-Hanke-Shankar in the cubic case
and of Siad in the monic case to binary forms of any degree with any fixed odd
leading coefficient. When $n$ is odd, we find that fixing the leading
coefficient increases the average $2$-torsion in the class group, relative to
the prediction of Cohen-Lenstra-Martinet-Malle. When $n$ is even, such
predictions are yet to be formulated; along with Siad's results in the monic
case, our theorems are the first of their kind to describe the average behavior
of the $p$-torsion in class groups of degree-$n$ fields where $p \mid n > 2$.
  To prove these theorems, we first answer a question of Ellenberg by
parametrizing square roots of the class of the inverse different of a ring cut
out by a binary form in terms of the integral orbits of a certain coregular
representation. This parametrization has a range of interesting applications,
from studying $2$-parts of class groups to studying $2$-Selmer groups of
hyperelliptic Jacobians.
