Yes! I hope I'm not the only one with a mode hopping HeNe. It's so mesmerizing, just watching the dot shift around, almost as if it is possessed. What kind of dot does your multimode GreeNe have?
More than likely when your 473 mode hops, it's hitting a slightly different area of the lens that isn't quite 'perfect'.. I have a slew of these.. both glass and acrylic.. that have spots in that that 'aren't quite right' But you can rotate the lenses a few degrees and the problem vanishes.