This talk exploits symmetry methods for the variational formulation of different problems in physics and chemistry. First, I will use symmetry methods to provide new variational principles for the description of mixed quantum states, in various pictures including Schrödinger, Heisenberg, Dirac (interaction) and Wigner-Moyal. Then, after discussing Ehrenfest's mean-field model, I will modify its symmetry properties to provide a new variational principle for expectation value dynamics in general situations. Upon moving to the Hamiltonian approach, this construction provides a complete dynamical splitting between expectation values and quantum deviations. As we shall see, specializing to Gaussian states yields energy conserving variants of previous models from the chemistry literature. In the last part of the talk, I will discuss some of the geometric features emerging in coupled classical-quantum dynamics.