The explanation of bar formation in the link wasn't very clear. I interpret him as saying that gravitational coupling of stars forces their orbits to be highly eccentric but precessing, with one focus at the galactic nucleus. These long, skinny ellipses are concentrated in the bars, giving the appearance of a rigidly rotating structure, although the stellar orbits are along the bar, not at right angles to it. This means that the two bars have independent populations, highly eccentric ellipses don't have foci in their centers, they have them at one end.

But are the bars themselves rotating about the galactic nucleus, or is this just what we expect to see? At the tip of each rigid bar is a graceful, curved "trailing" spiral arm, but this does not necessarily mean the bar is rotating, the spiral arm density wave could be a standing wave, or the galaxy could be rotating but the bar might not (the stars are only visible when they pass through as formation of short lived bright stars only occurs when the bar sweeps by, as it does in other density waves.

None of this is very clear, and I bet the astronomy is not settled either.
But one thing is clear, Even the simple laws of Newtonian orbital mechanics can give rise to highly varied and complex structures all sorts of secondary and tertiary resonances and harmonics occur, giving rise to totally unexpected patterns and configurations. There is much more to classical mechanics than "F = ma and you can't push a rope".