We introduce the classes of color blind and typhlotic cellular automata, that is, cellular automata that commute with all symbol permutations and all symbol mappings, respectively. We show that color blind cellular automata form a relatively large subclass of all cellular automata which contains an intrinsically universal automaton. On the other hand, we give simple characterizations for the color blind CA which are also group homomorphisms, and for general typhlotic CA, showing that both must be trivial in most cases. Using an extension lemma for a generalization of color blind CA, we show that the centralizer of a finite equicontinuous family of CA always contains an intrinsically universal automaton, as does that of an almost equicontinuous family with a blocking word of certain special type.