A pat is simply a wad of paper arising from any strip, straight or otherwise, which has been folded round and round; but in a consistent direction to keep it from springing apart when released. It is much more interesting for the pat to have a linear substructure, whereby it is perceived as a one-dimensional strip whose constitutents are similar wads. The simplest composite is a pair, and by taking a flexagon to be a pair of pats, the possibility exists to transfer one of the wads, or subpats, from one member of the pair to the other. In essence, that is the basis of Oakley and Wisner's definition of a flexagon.
Strictly, that is the definition of a triflexagon, because by letting one member of the overall pair be a triplet rather than a pair is the way to introduce tetraflexagons. Of course, to get a geometrically pleasing toy, all of this has to be realized via polygons fitting together around a common center giving a ring which may or may not want to lay flat.
Provided that the geometric constraints are met, there is no reason that the members of pairs cannot be triplets, or that the members of triplets cannot be pairs. That is particularly evident if one is willing to forego opening then up, and to regard a wad with internal structure as a monolithic block instead.
The simplest mixture has a map consisting of one triangle joined onto one square. The common meeting ground, if one were needed, would be a flexagon based on dodecagons, of which the simplest would be a ring of twelve.