We develop new mathematical results based on the spherical harmonic convolution framework for reﬂection from a curved surface. We derive novel identities, which are the angular frequency domain analogs to common spatial domain invariants such as reﬂectance ratios. They apply in a number of canonical cases, including single and multiple images of objects under the same and different lighting conditions. One important case we consider is two different glossy objects in two different lighting environments. Denote the spherical harmonic coefﬁcients by Blight,material lm , where the subscripts refer tothe spherical harmonic indices, and the superscripts to the lighting (1 or 2) and object or material (again 1 or 2). We derive a basic identity, B1,1 lm B2,2 lm = B1,2 lm B2,1 lm , independent of the speciﬁc lighting conﬁgurations or BRDFs. While this paper is primarily theoretical, it has the potential to lay the mathematical foundations for two important practical applications. First, we can develop more general algorithms for inverse rendering problems, which can directly relight and change material properties by transferring the BRDF or lighting from another object or illumination. Second, we can check the consistency of an image, to detect tampering or image splicing.