Conjugate interfaces are defined as pairs of interfaces between transparent media with the same Fresnel reflection coefficient for the parallel polarization (rp) at the same angle of incidence (φ). If ε′ denotes the ratio of dielectric constants of the media of incidence and refraction, respectively, we find that, ε′1+ ε′2= cosec2φ, for conjugate interfaces. Loci of points (ε′1, ε′2) that represent conjugate interfaces of the same reflection coefficientrpare shown to be a family of hyperbolas. We also define reciprocal interfaces as conjugate interfaces with ε′2= 1/ ε′1. The physical significance of conjugate interfaces is illustrated by examples.