Under what conditions is there a one-to-one mapping between continuous-time and discrete-time signals?
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?

Yes.
Say you're starting with $x(t)$, which is perfectly bandlimited to less than $f_B = \frac{1}{2 T_s}$, and you take samples at every $t = T_s k$: $x_k = x(T_s k)$.
All you need to do is to treat the  as a train of Dirac impulses (i.e., let $x_p(t) = \sum x_k \delta(t - T_s k)$), and then filter $x_p(t)$ with a brick-wall low-pass filter.  I.e., let $x_f(t) = h(x_p(t))$, where $h$ is that brick-wall filter.  $h$ will have an infinitely long impulse response, so you cannot realize this system in the real world - but not only will the mapping from $x_k$ to $x_f$ be exact, but if $h$ has zero phase delay then you'll find that $x_f(t) = x(t)$.