  The two-time nonequilibrium correlation and response functions in 1D kinetic
classical spin systems with non-conserved dynamics and quenched to their
zero-temperature critical point are studied. The exact solution of the kinetic
Ising model with Glauber dynamics for a wide class of initial states allows for
an explicit test of the universality of the non-equilibrium limit
fluctuation-dissipation ratio X_{\infty}. It is shown that the value of
X_{\infty} depends on whether the initial state has finitely many domain walls
or not and thus two distinct dynamic universality classes can be identified in
this model. Generic 1D kinetic spin systems with non-conserved dynamics fall
into the same universality classes as the kinetic Glauber-Ising model provided
the dynamics is invariant under the C-symmetry of simultaneous spin and
magnetic-field reversal. While C-symmetry is satisfied for magnetic systems, it
need not be for lattice gases which may therefore display hitherto unexplored
types of non-universal kinetics.
