The way in which a saxophone hangs on its sling ring is determined by the place of the ring in relation to the centre of gravity. The centre of gravity will always want to be in line with the sling. The natural position when playing the instrument makes this impossible: the thumbs steer the instrument to the correct position. But what remains is a force that tries to tilt the instrument. The direction of this force is determined by the place of the ring. When the ring is placed high it will make the instrument want to turn inwards at the mouthpiece side; when it is placed lower the instrument turns outwards.
Now what do we want?
An instrument that turns towards you at the mouthpiece side has a tendency to rest on the lower lip. But the lower lip has other business when playing the instrument. This makes that the left hand thumb must keep the instrument stable, even when activating the register key. But in the opposite situation, when the mouthpiece has a tendency to turn outward, the mouthpiece will rest against the upper teeth. The lip remains free for embouchure control and the left thumb is also free to handle the register key. Now it is the right thumb that controls the position of the instrument.