Representing the appearances of surfaces illuminated from different directions has long been an active research topic. While many representation methods have been proposed, the relationships between the different representations have been less well researched. These relationships are important, as they provide (a) an insight into the different capabilities of the surface representations, and (b) a means by which they may be converted to common computer graphics application formats. In this paper, we introduce a single mathematical framework and use it to express three commonly used surface texture relighting representations: Surface Gradients, Polynomial Texture Maps (PTM) and Eigen base images. The framework explicitly reveals the relations between the three methods, and from this we propose a set of conversion methods.