The amazing thing is that this is a scientifically productive rule—finding a new representation that gets rid of epiphenomenal distinctions, often means a substantially different theory of physics with experimental consequences!

This "Anti-Epiphenomenal Physics" is well known by a less fancy name of symmetry. Looking for hidden symmetries (and applying the Noether theorem to them, whenever possible) is the basic tool theorists use all the time. If anything, they go farther than that by reconstructing broken symmetries or even designing theories with hard-to-imagine symmetries.