How can a zero-padded length n signal be truncated to n coefficients from which a signal can be reconstructed?
From a theory standpoint, that ought to be possible. It's not bound to be simple, however. Zero padding will scale and interpolate the FD representation of your signal. If you remove coefficients the right way, you should be able to reconstruct what's missing.

Let's say you doubled the signal length by padding. You can then remove every other bin of the fourier transformed (starting with the second) and get exactly the result you would have without padding.