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:
http://www.geocities.com/CapeCanaveral/Hangar/7773/flexagon.html