There's an amazing way to derive Maxwell's equations, and the equations for 3 of the other fundamental forces of nature, directly from Lie group symmetry. You try to write down a theory that is symmetric under -local- symmetry transformations, meaning that the theory should give the same predictions even if you rotate (in some abstract space that you tack into the theory) by an angle that depends on position arbitrarily. At first this seems impossible, because any derivatives with respect to position will depend on the spatial variation of the rotation angle. But if you add an additional field that subtracts off the variation in the rotation angle, you find that this field is a dynamical object that coincides with the electromagnetic field! (Or to be more correct, it's the vector potential, which is directly related to electric and magnetic fields).
So there's some strange sense in which these laws of nature seem to arise from, or are at least deeply connected to geometry.
Indeed you can use symmetry, but it feels more like a mathematical hack, and the fact that it agrees with reality could be a coincidence. You can state that, and there is a lot of evidence for, that nature follows some basic geometrical rules. Applying that through a Lie theory framework on a symplectic manifold to see how charges behave differentially will eventually get you to Maxwell equations because of how those Lie algebras operate. However for me the real revelation was just using the Lienard–Wiechert approach to calculate how charged particles should behave in a relativistic field, which is as simple as it gets, and then see that you can build the full electromagnetic theory on top of that, with the bonus that the formulation is already relativistic. The same resulting symmetry in a corresponding Lie group is consequence of that (nicely captured by Hodge's equation), and invariance or operator rules don't need to be forced.
In the "opposite" direction, you might discover quantum mechanical "spin" from the Maxwell equation. Suggesting that coincidence is a kind of historical artifact :)
Thanks for the postclassical angle on this, I missed that in the comment below, which was only "charge"
Not sure what you mean by Hodge equation, care to elaborate?
I assume (for the lay physicist) it's the Hodge decomposition mentioned in here (pp6-8)
Correct, the famous d*F=J differential form formulation with one of the versions of the Hodge operator, which I have seen named in several ways. Also depending on your definition of the star operator and current density, you often see this as two equations with Hodge duals, like dF=0 plus d*F=*J. The tensor equivalent can be stated as a single equation or as a set, too.
To be fair and looking back at history, the discovery of Maxwell equations, relativity and quantum theory are so intertwined with the discovery, invention and application of new Mathematical ideas, in particular emanating from the work of Hamilton, Grassmann and then Lie, Levi-Civita, Cartan, etc. that is difficult to separate at what extent those concepts influenced over each other in their attempt to explain and describe reality. The ability to express Maxwell equations in a compact form with quaternions before vector calculus was even a thing provides some evidence. One can argue that the classical formulation for electromagnetism could be expressed that way because Hamilton was trying to find the proper framework that could capture his ideas about physics. Fast forward some 60 years and you also have a similar thing happening with Pauli matrices in quantum theory, and the work of Noether in modern physics.
Mixed Hodge structure en.wiki could use better (undergrad physics) examples if it wants to be the quaternion of our time, thank you for that rabbit hole :)
Somewhat related is noether's theorem (from Emmy Noether) that draws direct correspondence between symmetries and conserved quantities. E.g. conservation of linear momentum corresponds to a system that is invariant to translations. So you can find some of the fundamentals of a system by looking at symmetries and Lie groups/algebras give you tools to look at symmetries.
To add on to your mention of the rotation in abstract space , this is a local transformation of the electromagnetic potential. Not saying that "rotation" is a terrible thing to call it. it's just not usually thought of as a literal rotation. How about "twisting the potential"? Eg "twist" electric field into magnetic field? Rotation would connote that this is not 1D.
Some also think of this additional Lie as a ("central") extension of the Galilei group?
So there's some strange sense in which these laws of nature seem to arise from, or are at least deeply connected to geometry.