  A canonical formulation of coupled classical-quantum dynamics is presented.
The theory is named symmetric hybrid dynamics. It is proved that under some
general conditions its predictions are consistent with the full quantum ones.
Moreover symmetric hybrid dynamics displays a fully consistent canonical
structure. Namely, it is formulated over a Lie algebra of observables, time
evolution is unitary and the solution of a hybrid type Schr\"odinger equation.
A quantization prescription from classical mechanics to hybrid dynamics is
presented. The quantization map is a Lie algebra isomorphism. Finally some
possible applications of the theory are succinctly suggested.
