Caveat: I'm a layman not a physicist, and this may be more semantics than physics.
In a universe where time runs backwards (think of a movie run backwards, where a vase that is shattered on the floor assembles itself and jumps up to the countertop), is cause and effect also reversed, or is it invariant? After all, the assembled vase is the result of the action that got it there...
The answers already posted answer probably your question. I just will add a little about what physics says about time reversibility. It does say that macroscopically past and future are distinguishable, but microscopically they are almost the same, but also not identical.
You saw in the other answers that thermodynamics, the science of large numbers of particles that then must be described macroscopically, can distinguish between past and future because disorder (entropy in physics) goes up with time, for any irreversible process (of which there are many, macroscopically, such as your vase breaking). Too many pieces to put them back together.
Still, there is cause and effect, you just know which came first
Microscopically it is different. The physics for all particle and radiation interactions are also causal, but in this case they are all almost completely reversible. Almost in a minute. But ignoring the almost, that means that if you ran a movie of particle interactions forwards or backwards they both would look legitimate, and you could not distinguish past and future.
The alsmot comes about because time reversibility, called T symmetry, is not an exact symmetry of nature. The weak interaction in physics (One of the 4 forces of nature) is not T symmetric. We see some weak interactions with particles that do not go the same way forward and backward. The same is true with another so called discrete symmetry (discrete: only two ways it can be), the mirror symmetry, called the parity P. There are right handed interactions and left handed ones, and they are sometimes not the same. Finally there is a charge symmetry (positive and negative), called C, which is also sometimes broken. But CPT as the combination of the 3 seems to be always symmetric.
There still some uncertainty whether there might be some strong force interaction with also slightly breaks P and T, and why there is more matter than antimatter in the universe, which is probably due to some C, P and or T breaking early in the universe history.
But we do know that each of,those, and particularly for your question, T is not a symmetry of nature. Google and look at the Wikipedia article and others for time symmetry. See the wiki article at https://en.m.wikipedia.org/wiki/T-symmetry
If one considers only reversible processes or few particle physics (that is, in any case one can forget about the 2nd principle) the answer is definitely yes, the cause and effect is preserved. Classical mechanics (including relativity) as well as quantum mechanics are indeed time-reversal symmetric. At a mathematical level, that means that if you consider the substitution $t\to-t$ in your physics equations, the equations will not be modified.
However, in the real world where the 2nd principle (of thermodynamics) is valid, the time-reversal of our universe is not indiscernible from the original one. Quite the contrary. In fact, in this time-reversed universe the entropy will decrease, and this would mean that the time-reversed universe would look a lot different from the normal one. The 2nd principle is not time-reversal invariant. In the real world it is $$S_{t+\Delta t}\geq S_t$$ where $S_{t+\Delta t}$ and $S_t$ are the entropies at time $t+\Delta t$ and $t$. In the time-reversed universe the 2nd principle changes into $$S_{t+\Delta t}\leq S_t$$ That is, in our universe entropy increases, while in the time-reversed universe it decreases.
Therefore time-reversal symmetry is valid only for reversible processes. One may say that the 2nd principle breaks the time-reversal symmetry of the nature, or in simpler words, it creates a well-defined arrow of time which distinguishes the future from the past.
That means that in the reversed universe the cause follows the effect.
On an abstract level actual universe and the hypothetical one you suggest are objects of the same category. The same laws apply and causality is preserved.
If this was all that we consider, then if you could perceive universe from outside of the timeline, as a function of time, you couldn't tell in which way the time flows. You could even challenge the notion of time flow. Causality could be applied successfully both ways.
Now this bidirectional causality, assuming it's entirely deterministic, would mean that given any state of the world at some fixed point in time we could deduce the entire timeline.
Let's assume this is how timelines actually work - they are built from a starting point, according to rules, in both directions.
In the actual world, it seems at least, that starting point in time was the beginning. In a reversed universe, it would be somewhere in the future. Perhaps at the end - assuming that if we reversed everything at the moment of our universe conception, it would just cease to exist.
Now, since the fixed state can be any state, it doesn't have to be that very peculiar state at the beginning of our timeline or any other very distinct state. If the state is chaotic, then no characteristic of that trait, no statistical value calculated, would allow to differentiate between past and future. Whatever metamorphosis would happen in one direction, it would be exactly as likely to happen in the other.
However if the fixed state is very distinct, like in case of our timeline, we can intuitively understand, applying probability to an area that doesn't have anything to do with empirical notion of probability, that the fixed state is this peculiar, in our past, not any other we have left behind or are yet to experience. If we could create universe upon universe, in almost every universe that we created chaotically, both directions of time would be dominated by meaningless chaos, only in a very small minority the chaos would turn into an ordered world and then into an illusion of beginning at the end. While in almost every universe that spawns from a very specific starting scenario one or both directions (hard to tell if there is the other direction of time from big bang, and what happened there, perhaps there isn't one for some reason) would after limited time of order turn into that chaos without distinct features.
At least that's how I see it.
So the way we differentiate between future and past is that we notice certain traits of universe that seem non-random, and how they deteriorate. I think it's worth mentioning, how certain other traits seem to become less and less random with time for some time, and that we call evolution. But this process is "short" lived, so given long enough time we get to know which direction is the direction of deterioration.
In a chaotic universe we will see local spontaneous pockets of order arise. However if the universe is entirely chaotic or reached far enough from it's starting point in any direction to become so, those pockets will be exactly as likely to happen with flow as they are to happen against it.
