Reflection in a spherical mirror behaves like inversion in the sphere, which is the 3D equivalent to inversion in a circle.
2D Reflections in circles [i think, just play along] can be generalized to arbitrary 2D curves by finding the normal line(s) through the curve passing through the point we wish to reflect and inverting that point in the respective circles of curvature.
Can This method be generalized to reflection in surfaces in 3D?