  The Hamiltonian nature of the precessional dynamics of the classical
Heisenberg model leads to reciprocal interactions amongst the spins. Heisenberg
spins are reciprocal in nature. In this work, we study the dynamics of a
nonequilibrium classical spin chain in which the neighbours interact through a
purely non-reciprocal exchange coupling [EPL 60, 418 (2002)] which preserves
rotational symmetry. The resultant dynamics conserves neither magnetization nor
energy. We uncover other local conservation laws in their place in the extreme
case of a strictly antisymmetric coupling. We show numerically that the model
undergoes an analogue of thermalization. We present results on the presence of
conserved quantities, their diffusive spreading and a hydrodynamic picture, and
the nature of the decorrelation front upon adding an initial perturbation to
the system.
