contradict the correspondence between space and time that is built into most of present-day physics. So one might wonder how then it could be consistent with experiments that have been done in physics?

One possibility, illustrated in the pictures below, is to have a system that evolves in time according to explicit rules, but for these rules to have built into them a symmetry between space and time.

Examples of one-dimensional cellular automata which exhibit a symmetry between space and time. Each picture can be generated by starting from initial conditions at the top, and then just evolving down the page repeatedly applying the cellular automaton rule. The particular rules shown are reversible second-order ones with numbers 90R and 150R.