Is the universe deterministic when looking backwards? Someone once told me that if, in theory, we could know the position and motion of all the particles in the universe, we could use that data to run time backwards, and work out everything that had come before.  That means the current state of the universe effectively encodes all of history.
Is this theory reasonable?
I have two main concerns, so it would be great if you could address these in your answer:


*

*Due to entropy, information is lost over time.  I believe that means the same state could be arrived at by two different histories.  (For example, once a glass of water is still or frozen, we don't know if it was poured 100 years ago or 10,000 years ago.)  If that is true, I don't see how we could determine history from state alone.

*From A Brief History of Time (Stephen Hawking) I got the impression that time was symmetrical: it acted the same regardless of which way it was running.  So if the universe is deterministic going forwards, that would also mean it is deterministic going backwards.  Was Hawking correct about time symmetry, or did I just misunderstand him?
 A: Two things to think about. Is physics (i.e., the description we understand of the universe) deterministic, and then is is time symmetric? The answer to the first is yes, and the second is no. This answers treats both because tHey are Intrinsically related to the question of whether the universe would evolve back to the same if we reversed time. Clearly, if it was time symmetric but not deterministic, nothing would ever go back to the same. 
This covers it all, though some of it summarily. But I try to describe and answer this using physics, and identify what may not be well known or accepted physics. There is lots of good physics to cover. 
The basic answer seems to be that it is deterministic (in a strange but well understood way even in quantum physics), but not time symmetric (both microscopically and macroscopically).
At its basics the universe is deterministic, though in practice that's a different question. The question of whether the universe is deterministic was asked and had a number of answers in 2013, see Is the universe fundamentally deterministic?
The answer that emerges there, and the answer backed by known physics, is that basically it is deterministic, if we believe the currently best known physics, the Standard Model and General Relativity (GR). That is at a microscopic level plus relativity. It ignores the possibility that unknown physics such as Quantum Gravity would say otherwise, since we don't have a theory for Quantum Gravity yet. Clearly it also ignores what may be found at much higher energies than we have been able to measure things at, in what would be called Beyond the Standard Model'. Also, GR may allow closed timelike curves (CTC), which would also imply a breakdown of causality (which requires nothing faster than light, in the physics we know). For causality reasons most physicists think CRCs are not possible, except in regions separated from us by event horizons such as inside black holes so that we would not be affected. There are some strange GR solutions that allow CRCs, but they seem to not be physically possible. That does remain a controversy. This is not philosophy, it is pure physics, and with that possible exception the universe is causal and deterministic (I know, an exception is an exception, we just don't know about those yet)  
There is another factor to account for, and two items to explain. First, account for entropy and thermodynamics, then about wave function collapse and the measurement problem. Both of those are still a bit controversial, but there is some semi-consensus that is emerging. That is the next two paragraphs. The third is simply a misunderstanding by laymen that quantum physics is not classically deterministic i.e., the uncertainty principle. That also is explainable and physicists agree that quantum physics is deterministic, eg, the Schrodinger, Dirac, and all other quantum physics (in quantum mechanics and in quantum field theories) all predict the quantum physics we see; the simplest way to see it's to understand that in QM the wave function is perfectly predictable. @Bush explains below that it is not if position or momentum is predictable but whether the wavefucntion (or other quantum equivalents) are deterministic. They are.   In overall quantum physics that comes from the fact that all the evolution equations we know for the standard model are unitary. That's the technical term for information is conserved. Black Hole physics seems to contradict it with the horizon, but even Hawking has agreed that information is conserved, and people are trying to solve the 'paradox'. See Hawking's latest described simply, and the reference to his June Phys Rev Letters at http://phys.org/news/2016-06-hawking-team-soft-hair-theory.html
For entropy and thermodynamics it's simpler. Those are simply statistical macroscopic observation we have to make when detailing the evolution of all the quantum fields everywhere in the universe can not be computed. It's a practical and smart way to deal with the largeness, but it is our lack of knowledge we compute in entropy, not the universe.
As for wave function collapse not being unitary, or breaking the quantum physics laws, the well understood (not by laymen) answer is that if you include the evolution of the measuring apparatus, it is unitary and there is no collapse, just interactions that make the original wave function decohere. It appears to collapse if we caused it, but it simply interacted with us unitarily.
So the symmetry going back in time can ignore all of that, and and simply be answered by whether quantum physics is time symmetric, plus the additional issue of initial conditions in the large. On time symmetry, it is known that physics is not. The weak force breaks time symmetry, and CP symmetry. The CP symmetry breaking has been known, it is why there are only left handed neutrinos. (CP symmetry may also be broken, very very weakly, by the strong force, there are physical observations that hint at it, but it is not clear, and it has never been measured to be so). CPT symmetry (T is time symmetry) has never been found to be broken, so when CP is broken so is T. It is complicated but it seems to be so, and it is hoped that the CP breaking will explain why there are more particles than antiparticles in the universe, still an unsolved issue. See on T symmetry at https://en.m.wikipedia.org/wiki/T-symmetry and a lot of references there and elsewhere. The weak force is not time symmetric.
The final argument is the question of why are we evolving forward? Macroscopically it is thought now that it has to do with the initial conditions of the universe and entropy. That it was created in a low entropy state, and the evolution microscopically to a state of higher and higher entropy is the reason for the direction of the time arrow. Well, macroscopically it seems reasonable, but is that inherent in the universe or our perception? Physically there is still arguments going back and forth and it is sometimes described as philosophy. The time arrow exists in microscopic physics in the weak force, but is is totally unclear how that manifests itself in macroscopic physics and entropy. 
But taking that into account then the answer is that if you run the universe backwards from where it is right now somerhing would be different. Microscopically, the weak force time asymmetry would change some of the evolution, and macroscopically the initial conditions would make it go towards a higher entropy rather to back to a small entropy. 
Lots of physics in that (which has physical, real effects) but some unknowns people call philosophy, and which may or may not have physical effects. 
A: Consider an arbitrary superposition state with n eigenstates. In our perception of time, let's call this 'forward time', when that superposition state is measured, it collapses to an eigenstate. Which eigenstate is chosen may or may not be determined by the measurement process. But consider this: the superposition told you a lot about the system it was in because it was a linear combination of n eigenstates, but after measurement, the quantum state consists of only one eigenstate, which could be the eigenstate of any number of systems.
Now let's consider this process through 'backward time'. You start with a quantum state that consists of one eigenstate. Is there enough information for this quantum state to transform into the arbitrary quantum state mentioned previously? All the literature I've read suggests no, there is not enough information. A quantum measurement is time asymmetric for this reason. 
This problem is rather deep and not fully understood at this time. I don't remember Hawking addressing this problem in his book, and this is a serious issue that any determinism proponent must address. Looking at the conclusion of A Brief History of Time, Hawking says:

The unpredictable, random element comes in only when we try to
  interpret  the wave in terms of the positions and velocities of
  particles. But maybe that is our mistake...

Here he addresses the effect of the Uncertainty Principle on determinism. I'd say he certainly leaves determinism up for debate, even without addressing the quantum measurement problem. 
To make the point of my answer clear: you based your question on the assumption that the universe is deterministic going 'forward' in time, but that is not an assumption that can be made. Considering there are time-asymmetric processes at the elementary particle level, whether the universe in 'backwards' time is deterministic will be an even more difficult question to answer. 
A: No, the universe is not deterministic when looking backwards.   The determination of position and velocity of a particle can NEVER (by the Heisenberg uncertainty principle) be known at any time, so neither forward-time nor backward-time
"predictions" of a particle trajectory are absolute.   
If you can't absolutely predict a particle, a universe is out of the
question.
