Suppose you have two rings of equal radius exactly overlaid on each other. One of them has uniform charge density $+\lambda$ and other uniform charge density $-\lambda$. Clearly the charges will simply cancel and there will be no electric or magnetic fields anywhere.
Now suppose you start the positive ring rotating in place at relativistic speed. There will now be current flow and thus a magnetic field, but for simplicity I'm going to ignore that and just consider the Lorentz force on a charged particle $q$ at rest with respect to the non-spinning ring. Naively, I might expect the positive ring to get Lorentz-contracted and therefore appear to increase its linear charge density, thus creating a net outward electric field at $q$ and repelling it. But this can't be right, because the total charge across any fixed constant-time slice is both conserved and Lorentz-invariant, so it must stay zero. Why doesn't the positive ring Lorentz-contract and appear to gain charge and repel the charge $q$?