We prove that, upon passing to nilreductions, there exists a unique rigid analytic map from the author's (cuspidal) half-integral weight eigencurve to its integral weight counterpart that interpolates the classical Shimura lifting. Up to the action of a canonical involution on the half-integral weight eigencurve, this map is shown to identify the cupsidal half-integral weight eigencurve with a union of irreducible components of the integral weight eigencurve.