A handwavy argument for this is that if you continuously deform your picture, you can bring an upright and an inverted image toward each other until they cancel

A handwavy argument for this is that if you continuously deform your picture, you can bring an upright and an inverted image toward each other until they cancel out (if you've ever played around with curved mirrors you probably can imagine how that would look). Eventually, after doing enough of these cancelations, you're left with an undistorted picture with a single upright image -- so, you must have had n inverted images and n+1 upright ones. The morse theory the wikipedia article mentions is roughly about being able to do this argument rigorously.