I started a blow-up on twitter about electromagnetism and pseudo-vectors. Why do we need to invoke the pseudo-vector magnetic field when we start with real vectors and end with real vectors? This is all related to my project with Soledad Villar (JHU) and Ben Blum-Smith (NYU) about universally approximating functions (machine learning) for physics. Kate Storey-Fisher (NYU) converted an electromagnetic expression (for that paper/project) that contains cross products and B field into one that requires no cross products and no B field. So why do we need the B field again?
I figured out the answer today: If we want electromagnetism to be a field theory in which charges create or propagate a field and a test particle obtains an electromagnetic force by interaction with that field, then the field has to be an order-2 tensor or contain a pseudo-vector. That is, you need tensor objects to encode the configuration and motions of the distant charges. If you don't need your theory to be a field theory, you can get away without the high-order or pseudo- objects. This should probably be on my teaching blog, not here!