How does counter-canter prepare a horse for flying changes in dressage?

Counter-canter — cantering on the outside lead through turns and curves where the horse's natural preference would be to canter on the inside lead — is the most important specific preparation for flying changes because it develops precisely the qualities that flying changes require: straightness through the horse's body, balance and collection in the canter, and the horse's ability to maintain a canter lead against its natural inclination without breaking. A horse that can hold good counter-canter through circles, serpentines, and corners without losing rhythm, breaking to trot, or spontaneously changing leads has demonstrated the physical and mental qualities that flying changes require — and a horse that cannot do this is not ready for changes regardless of other training achievements. Counter-canter develops straightness by requiring the horse to maintain alignment through its body while cantering on the outside lead, which demands that the horse not throw its weight inward or outward to compensate for the outside lead but instead carry itself with genuine straightness. This straightness is exactly what flying changes require: a horse that swings its haunches or drifts its shoulders to facilitate a change is lacking the straightness that good counter-canter develops. Counter-canter also develops collection in the canter: maintaining balance on the outside lead through turns requires the horse to carry more weight on its hindquarters than simple straight-ahead cantering demands, gradually developing the collecting capacity that the suspension required for clean changes requires. The psychological preparation counter-canter provides is equally important: a horse that waits patiently in counter-canter without anticipating a change has learned to wait for the rider's aids — the most fundamental behavioral prerequisite for flying changes that come exactly when asked rather than when the horse decides the moment is appropriate.