I'll concentrate on cellular automata in this answer, because it's a good example, and should help to give a good intuition about algorithms in general.
The answer is: most cellular automata do have an intrinsic time direction, but some don't.
The most famous example of a cellular automaton is John Conway's Game of Life. This is an irreversible cellular automaton: it has a definite direction of time, in the sense that you can't run it backwards. Running it forward is easy: every cell gets turned on or off according to definite rules that depend on its state and the state of its neighbours. But in reverse this doesn't always work. For example, suppose that at time $t$, some particular cell is off, and all its neighbours are also off. Was this cell on or off at time $t-1$? There is no way to know. If it had been on in the last time step, it would be off now, because all its neighbours are off - but if it had been off then, it would also still be off now. So there's no reverse version of the algorithm. It loses information as it runs.
But this is not the case for all cellular automata. There are "reversible" ones, where you can always work out the previous state. My favorite example is called Critters, although there isn't much information about it online. (If you're prepared for an in-depth read, there's a nice paper that describes it.) Reversible cellular automata have some interesting properties. For example, if two gliders collide, at least one glider must emerge from the resulting debris. If it didn't, the debris would eventually go into a cycle (exactly repeating one of its previous states) and then you'd lose the information about when the glider collision occurred. Since a reversible cellular automata can be run "backwards" from its final state to its initial state, there's a sense in which it doesn't have an intrinsic direction of time.
There are analogues of all this in the wider world of algorithms. There are a lot of known techniques for constructing reversible algorithms, collectively known as reversible computing. This is quite an important area of computational theory, because the theory of quantum computing builds upon it.