[adrian_b]

You think this way because you do not pair in the right way the Maxwell equations. The divergence equations are not paired. The same for the curl equations. Pairing the equations in the wrong way can actually lead to errors in the relativistic case.

Two equations are intrinsic properties of the electromagnetic field because it is derived from a potential, i.e. the null condition for the divergence of the magnetic field and Faraday's law of induction.

The other two equations are what you call "physical", i.e. they show the relationship between the electromagnetic field and its sources, i.e. electric current and electric charge.

Alternatively, if you use potentials to describe the electromagnetic field, which is better in my opinion, you just have the relations between potentials and their sources (together with the conservation law for the electric charge). With potentials, you can get rid of the "geometrical" relations (though the choice of potentials is not unique).