I think it is unlikely that you could get a complete reflection from ice as it is formed of separate crystals, each of which has its own refractive index, which I believe varies depending on the alignment of the individual crystals, ice which has been compressed, which would be the case where ice caves form, will have the crystals further deformed, so any light entering the matrix of crystals would be scattered.
I believe the best chance of a reflection in an ice cave would be in the thin layer of liquid water which would line the wall of the cave, as it forms, this would have a different refractive index to the ice, so there is a possibility that a reflecion could form at the interface, but as light passes through both water, and ice, it would not be a complete reflection.