  Einstein-aether theory is extended by allowing for spinning degrees of
freedom of the aether. In addition to the acceleration, shear, expansion, and
vorticity of the aether velocity field, a spin rotation describing the dynamics
of a classical intrinsic angular momentum of the aether is introduced as a
kinematic quantity. The action of Einstein-aether theory is augmented by a term
quadratic in the spin rotation and by coupling terms with the vorticity and the
acceleration. Besides breaking the Lorentz boost invariance, the theory breaks
the invariance under spatial rotations in the direction of the aether velocity.
In the weak field limit, there is a linear relationship between the spin
rotation, the vorticity, and the acceleration. Linearized wave solutions
correspond to the ones of Einstein-aether theory where the speeds of the spin 0
and spin 1 mode are modified. The extension of Einstein-aether theory has a
natural formulation in the framework of a teleparallel geometry where the
kinematic quantities become torsion fields.
