These are the first thoughts on how to represent Wavefunctions.
There are two main basis set types: atom centered and absolute
Atomic basis set can always be represented as a radial function and an l value that specifies which spherical harmonics $Y_m^l$ are used for the radial part. For the radial part one could store just numerical values on a logarithmic grid and still have enough accuracy. This is natural for fhi-aims. As long as no mixed basis set are used I think fhi aims integrals can be used to calculate any derived quantity wanted.
Plane waves are normally defined through a cutoff, but in practice FFT have to be calculated on a real space grid of a given size. This size should be stored. Real space grid, keep all coefficients, while plane waves normally in the reciprocal space use a spherical cutoff.
While one can try to recover the core electrons from a pseudo potential calculation (as it is done for NMR) in general it is difficult.
In approaches like GAPW the density in pane waves can be seen as a projection of the total density, which shows that the inversion (plane waves -> all atoms in general might need a self consistent optimization. It is possible that for given pseudopotentials this might be simpler (especially if the projectors are known).
The fitting of pseudopotentials using scalar relativistic solutions (so that the unrelativistic solution is close to the relativistic one) murkies further the things.
Wavelets can be refined close to the atom, but such basis set are not considered.
If Numerical orbitals are used for all representations some approach migtht be envisaged (using the potential values as stating point?).