The beam is definitely multimode and not a clean TEM00.
It looks like the mode-hopping HeNe daguin had a while ago. The dot stayed round, while the interior went crazy.
There's not much you can do about it, and it's most likely caused by imperfections within the crystals.
Having said that, though, the beam still stays in one piece, and when beam-pointing, it'd be barely noticable.