I believe inverting the bottles is unnecessary, the yeast will do it's work regardless; keep in mind the yeast actively working on your priming sugar is mostly the one suspended in the beer (which it's always there in a greater or lesser degree, unless the beer is filtered and/or pasteurized), not just the one that settles at the bottom. Not that anything bad would happen; the yeast readily consume any oxygen originally left in the headspace (assuming you didn't cap on foam or flushed the bottle with CO2 prior to filling), the CO2 produced will certainly push it into the beer anyways, and yeast love oxygen. It is my understanding that some Belgian producers condition their bottles lying on the side, which makes sense considering the contact area between the beer and yeast is greater this way.