When an E20/30 is switched off, it (over)writes a single-point gpx file with the current location. (The file is called .Position.gpx note the initial dot)
I've no idea what use this is supposed to be, since if it's hot-starting or warm-starting it certainly doesn't need this information, and if it's cold-starting the chances are it'll be somewhere else anyway. But presumably it reads this file on startup and if it's been moved in the interim that may be why the effect John describes happens - though obviously it shouldn't, and it doesn't explain what ocurred in the exact case above where the GPS wasn't moved.