According to some (e.g. Haroche and Raimond in Exploring the quantum: atoms, cavities and photons), the quantum world consists (mainly) of spins and harmonic oscillators.
For harmonic oscillators (i.e. bosons), it is well known that they can be appropriately described in $(x,p)$ phase space, which satisfies a 'symplectic' structure (see e.g. Gaussian states in continuous variable quantum information). A system with coupled bosonic modes can be symplectically diagonalized into the eigenmodes.
My question is: is there a similar structure for spin states, living in $(\sigma^x,\sigma^y,\sigma^z)$ space (without resorting to a Holstein transformation or similar)? For simplicity, I'm mainly interested in the spin-1/2 case.