The definition of causal invariance originally used by S. Wolfram is the following.
[A] substitution system […] has the property of causal invariance
[if and only if] it gives the same causal network independent of the scheme used
to apply its underlying rules.
It is interesting to contrast it with Jonathan’s definition.
Definition 22. A multiway system is “causal invariant” if and only if
the causal networks associated with all paths through the multiway
system are (eventually) isomorphic as directed, acyclic graphs.
Here is a previous discussion about this subject.