  The damage spreading (DS) transitions of two one-dimensional stochastic
cellular automata suggested by Grassberger (A and B) and the kinetic Ising
model of Menyh\'ard (NEKIM) have been investigated on the level of kinks and
spins. On the level of spins the parity conservation is not satisfied and
therefore studying these models provides a convenient tool to understand the
dependence of DS properties on symmetries. For the model B the critical point
and the DS transition point is well separated and directed percolation damage
spreading transition universality was found for spin damage as well as for kink
damage in spite of the conservation of damage variables modulo 2 in the latter
case. For the A stochastic cellular automaton, and the NEKIM model the two
transition points coincide with drastic effects on the damage of spin and kink
variables showing different time dependent behaviours. While the kink DS
transition is continuous and shows regular PC class universality, the spin
damage exhibits a discontinuous phase transition with compact clusters and PC
like dynamical scaling ($\eta^,$), ($\delta_s$) and ($z_s$) exponents whereas
the static exponents determined by FSS are consistent with that of the spins of
the NEKIM model at the PC transition point. The generalised hyper-scaling law
is satisfied.
