StartsWithABang writes: One of the most important characteristics of a planet, at least according to the IAU definition, is that it clear its orbit of all other bodies. But if we allowed for a special caveat — the possibility of two similarly-sized objects sharing the same orbit — could we have a stable configuration where that occurred? Surprisingly, not only is the answer yes, but there are three ways to do it: to have one at the L4/L5 Lagrange point of the other, to have a close-orbiting binary planet, or to have orbit-swapping worlds, where they periodically change spots with one another. Unbelievably, our Solar System has a history of all three!
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