  In this paper we study the space evolution in the Rule 54 reversible cellular
automaton, which is a paradigmatic example of a deterministic interacting
lattice gas. We show that the spatial translation of time configurations of the
automaton is given in terms of local deterministic maps with the support that
is small but bigger than that of the time evolution. The model is thus an
example of space-time dual reversible cellular automaton, i.e. its dual is also
(in general different) reversible cellular automaton. We provide two equivalent
interpretations of the result; the first one relies on the dynamics of
quasi-particles and follows from an exhaustive check of all the relevant time
configurations, while the second one relies on purely algebraic considerations
based on the circuit representation of the dynamics. Additionally, we use the
properties of the local space evolution maps to provide an alternative
derivation of the matrix product representation of multi-time correlation
functions of local observables positioned at the same spatial coordinate.
