The fundamental problem with transforming normals is largely a product of our mental model of what a normal really is. A normal is not a geometric property relating to points of the surface, like a quill on a porcupine. Instead normals represent geometric properties on the surface. They are an implicit representation of the tangent space of the surface at a point.

In three dimensions the tangent space at a point is a plane. A plane can be represented by either two basis vectors, but such a representation is not unique. The set of vectors orthogonal to such a plane is, however unique and this vector is what we use to represent the tangent space, and we call it a normal.