If you look you will see a finger that locates the cutter. The tool above looks like it could be some form of a locating finger for a cutter grinder. If your parting with it let me know as I'm in need of one and I could make that work.
Ok, so the strap is meant to be flexible enough to rotate the cutting tool your sharpening in one direction, but stiff enough to act like an index in the other direction?