We have identified ultracold atoms in magneto-optical double-well potentials as a very clean setting in which to study the quantum and classical dynamics of a nonlinear system with multiple degrees of freedom. In this system, entanglement at the quantum level and chaos at the classical level arise from nonseparable couplings between the atomic spin and its center of mass motion. The main features of the chaotic dynamics are analyzed using action-angle variables and Poincaré surfaces of section. We show that for the initial state prepared in current experiments [D. J. Haycock et al., Phys. Rev. Lett. 85, 3365 (2000)], classical and quantum expectation values diverge after a finite time, and the observed experimental dynamics is consistent with quantum-mechanical predictions. Furthermore, the motion corresponds to tunneling through a dynamical potential barrier. The coupling between the spin and the motional subsystems, which are very different in nature from one another, leads to interesting questions regarding the transition from regular quantum dynamics to chaotic classical motion.