Required reading: trihexaflexagon

The hexahexaflexagon looks pretty much like the trihexaflexagon, except it has no visible folds; it is all edges (foldable edges, mind you). A hexahexaflexagon is made by taking a strip of paper, turning it into a flattenned spiral, then turning that into a trihexaflexagon (the trihexaflexagon is a flattenned moebius strip).

The hexahexaflexagon can be flexed with more options than a trihexaflexagon, allowing a total of 6 faces. It has 3 faces which are more common than the others, as can be seen in a Tuckerman traversal. Also, there is a way to screw up your hexahexaflexagon, so that the triangles on the faces will be mixed up between faces, and the topology of the thing becomes irregular; I will be covering this topic when I figure out how to express it in text.

Again, for construction details, I refer you to:

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