The double Lorentzian symmetries of the phase space kinematics of a hyperbolic position space in special relativity imply the existence of separate rotation subgroups in position and velocity space. In velocity space this is the angular momentum of the Thomas precession; a similar term appears in curved position space. These recouple to form a standard total angular momentum J, together with an unfamiliar companion contra-angular momentum Q. Its properties, Lie algebra and quantization are developed.