The ability for classically trained dancers to hold "up" their bodies and get on their alignment axis (and turn/spin without dizziness) gives them a huge advantage over non-dancers.
Dizziness is a psychological response to physiological stimuli. The fluid in your ear will spin regardless of whether one spots or shapes a turn, no matter if you are on your axis or not, one will always be "dizzy". You cannot stop the fluid from moving when you move.
But you can turn off the psychological reaction, simply through repetition, in the end cutting off the brains automatic assumption that the body is in danger and needs to sit down for safety's sake. It is repetition and mental focus that dampens the panic response and allows the body to continue to function in a normal manner.
You also have receptors in your feet that tell you whether or not you are standing up or falling sideways, as well as proprioception coming in from all parts of the body. Training your brain to pay attention to those signals, instead of the spinning fluid in the loops in your ear, is the key to overcoming dizziness.