Addresses certain questions concerning invertible cellular automata, and presents new results in this area. Specifically, we explicitly construct a cellular automaton in a class (a residual class) previously known not to be empty only via a nonconstructive existence proof. This class contains cellular automata that are invertible on every finite support but not on an infinite lattice. Moreover, we show a class that contains invertible cellular automata having bounded neighborhood, but whose inverses constitute a class of cellular automata for which there isn't any recursive function bounding all the neighborhood.< >