Beyond the established symmetries of physics - Charge, Spin, Color, Time, etc. - is there any such thing as symmetry of Information?

One clarification: electric charge, angular momentum, and color charge are conserved quantities, not symmetries. Time is a continuous symmetry though, and its associated conserved quantity is energy.

Similarly, information isn't a symmetry, but it is a conserved quantity. So I assume you're asking if there's an associated symmetry for it from Noether's theorem. This is an interesting question: while Noether's theorem ensures that any continuous symmetry will have a corresponding conserved quantity, the reverse isn't necessarily true as far as I know. In the case of information conservation, this normally follows naturally from the fact that the laws of physics are deterministic and reversible (Newton's laws or the Schrodinger equation).

If you insist on trying to find such a symmetry, then you can do so by equating conservation of information with the conservation of probability current in quantum mechanics. This then becomes a math problem: is there a transformation of the quantum mechanical wavefunction (psi) that leaves its action invariant? It turns there is: the transformation psi -> exp(i*theta)*psi. So it seems the symmetry of the wavefunction with respect to complex phase necessitates the conservation of probability current (i.e. information).

Edit: Looking into it a bit more, Noether's theorem does work both ways. Also, the Wikipedia page outlines this invariance of the wavefunction with complex phase. In that article, they use it to show conservation of electric current density by multiplying the wavefunction by the particle's charge, but it seems to me the first thing it shows is conservation of probability current density. If you're interested in other conserved quantities and their associated symmetries, there's a nice table on Wikipedia that summarizes them.