  We develop a microscopic theory of spin-lattice interactions in magnetic
insulators, separating rigid-body rotations and the internal angular momentum,
or spin, of the phonons, while conserving the total angular momentum. In the
low-energy limit, the microscopic couplings are mapped onto experimentally
accessible magnetoelastic constants. We show that the transient phonon spin
contribution of the excited system can dominate over the magnon spin, leading
to nontrivial Einstein-de Haas physics.
