Every edge labelled with a particular letter has the same orientation; each rhombus is kind of like a commutative-diagram 2-cell asserting the commutativity of two letters. The "concurrent equality" says that you should consider equal the two otherwise different ways of rewriting abcde to baced; the one that transposes ab first, and the other that transposes de first. Since they don't interfere, they're equal.
One cute thing in the case of n=5 is that the rhombuses involved are Penrose tiles. Dunno if this has anything to do with anything.
Running some experiments, it's clear that this sequence counts the number of ways to transpositionally invert a string, without the crazy concurrent equality — which I guess corresponds to some sort of ordered rhombus-tiling of a 2n-gon.