Apart from practical errors related to the truncation of the infinitely-long energy signal when discretized to digital, can we find a UNIQUE mapping between the two signal forms?
I think about this differently than Tim Wescott's in his excellent answer and I hope we can make a fun discussion out of this. Tim accurately describes that any bandlimited signal can be sampled without any loss of information and can be reconstructed into the original analog signal again.
So where is the rub? Turns out bandlimited signals do NOT exist. In order to be bandlimited a signal must be infinitely long in time. That's not possible. The same goes for the reconstruction filter: not only does it require infinite time, it's also infinitely non-causal: you have to go backwards in time past the big band to "correctly" apply.
My point here is that dealing with finite sequences isn't just a "practical" problem, it's a fundamental one. A unique mapping does exist in theory but only under assumptions which are ALWAYS guaranteed to be violated. Whether that qualifies are not, really depends on what exactly do you mean by "exist".
Here is an interesting example: let's say you have an signal that's sampled at 44.1kHz. You ought be to be able to sample rate convert this to 48 kHz and then back to 44.1kHz getting back the original samples. That's possible in "theory" but it's impossible to actually build a contraption that can do this.