I have not yet taken the time to try and unravel the secrets of pure 22.5 degree angle origami design and, consequently, don't have a very thorough understanding of how that set of design tools can be used. However, I have discovered an exceptionally useful variation on 22.5 degree angles in the works of Kota Imai that I have diagrammed below. The diagrams only cover the basic concept, but I have included one of my own CPs with its hinge polygons outlined to help enforce a basic understanding of how this method can be applied. At first the idea itself may seem a bit complex, but with time and practice, it will start to make sense, and, once it does, may prove to be extremely useful.
The CP for my Flying Acrophylla Titans (stick insect).
This a very complex crease pattern, but I feel it effectively demonstrates the the application of offset 22.5 degree angles in many different situations. Note that the stretch going between the front legs (the pair of large, pleat laden points at the top) and wings (two smaller flaps side by side in the upper middle) is a perfect offset pythagorean stretch and different than the offset 22.5 angles throughout the rest of the CP. For more examples I highly recommend visiting Kota Imai's Flickr page.