  We generalize a classical reciprocity law due to R\'edei using our recently
developed description of the $2$-torsion of class groups of multiquadratic
fields. This result is then used to prove a variety of new reflection
principles for class groups, one of which involves a symbol similar to the spin
symbol as defined in work of Friedlander, Iwaniec, Mazur and Rubin. We combine
these reflection principles with Smith's techniques to prove Stevenhagen's
conjecture on the solubility of the negative Pell equation.
