To the extent that it's still valid (anaesthia vs actual sleep) and the results also apply to humans (it was in mice - but if you look at a family tree, rodents are the closest mammals to primates - which is us), there was at least one study showing that sleep position could affect the efficiency of glymphatic clearance; lateral (side) worked better than prone (back) or supine (stomach) 'sleeping' (in quotes because it was anaesthia not actual fall-asleep sleep). I'd add a link but I'm on a mobile device and my notes are at home. Sorry!
That said, as far as I know, we *still* don't really know how general anasthetics work. They seem to have impairing the electron transport chain in common, but how that translates into loss of consciousness...(?) I suspect we won't know until there's better understanding of what consciousness *is*.