In Que Será Serees, Seeres shares (or nearly shares) an orbit with its sister world Bailera. Every 201.5-ish Serees/Bailera years (about 193-2/3 Earth years), they closely approach each other, switch orbits, and then retreat from each other. I’m sure some have wondered if this is possible.
The basis of this pair comes from the Saturnian moons Janus and Epimetheus. These moons share the same average orbital distance from Saturn — average, but not exact. Their orbits vary just enough that when they approach they each affect the others orbital velocity and effectively swap orbits (link).
Well, that’s fine for moons, but what about planets? While we haven’t found any co-orbiting planets (yet), we have found a pair that hint at what the folks on Serees see during a Bailariat: Odd Alien Planets So Close Together They See ‘Planetrise’.
I could also show you the pages of orbital mechanics I calculated when I was in the preparatory world-building stage before I started writing the novel, but I somehow think that would be overkill. Besides, the Serees/Bailera pair is just hypothetical, after all.