Looking-Glass Relativity: Orientation-Gauge Geometry and a Symmetric Extension of the Equivalence Principle

The analytic Kerr continuation across its regular central disk yields two asymptotic descriptions whose masses have opposite signs relative to a single continued stationary generator. We seek a completion in which neither description is privileged: transport may reverse the local choice of future, observers adapted to either choice formulate equivalent native laws, and the Einstein–geodesic system is recovered exactly whenever only one orientation is populated. We prove that a single Lorentzian metric cannot support two distinct geodesic free-fall structures and that an ordinary unconstrained local action cannot preserve strictly native matter equations while generating the required opposite gravitational source terms. The resulting framework assigns an exchange-symmetric metric datum with two Lorentzian components to a single physical spacetime, each component governing ordinary free fall for one matter orientation. We rederive the four signed-mass interaction classes, establish neutral-pair conservation, and formulate a local nonlinear projected variational principle whose Bianchi identities determine the comparison equations. The vacuum quadratic sector contains two helicity-two degrees of freedom per metric component. The weak-field limit reproduces the complete four-class interaction law for arbitrary mixtures, while the reciprocal Kerr pair removes the regular-disk Israel layer. A sufficient nonlinear constraint criterion and the compact ring-core problem are stated separately.