Abstract Cellular automata find applications in cryptography, error correction, theoretical computer science, and the simulation of physical and biological processes. We say that a Boolean function is a proper lifting if its induced cellular automaton is reversible. This paper introduces new classes of proper liftings, constructed through compositions of landscape functions, and explores their cryptographic potential by computing indicators relevant to differential and linear cryptanalysis. We combine theoretical and computational methods to obtain a complete classification of proper liftings for diameters up to 6. Several open problems and conjectures are posed along the way.