  We develop and apply a new framework for reconstructing the ionization
history during the epoch of recombination with combinations of cosmic microwave
background (CMB), baryon acoustic oscillation (BAO) and supernova data. We find
a wide range of ionization histories that are consistent with current CMB data,
and also that cosmological parameter constraints are significantly weakened
once freedom in recombination is introduced. BAO data partially break the
degeneracy between cosmological parameters and the recombination model, and are
therefore important in these reconstructions. The 95% confidence upper limits
on H0 are 80.1 (70.7) km/s/Mpc given CMB (CMB+BAO) data, assuming no other
changes are made to the standard cosmological model. Including
Cepheid-calibrated supernova data in the analysis drives a preference for
non-standard recombination histories with visibility functions that peak early
and exhibit appreciable skewness. Forthcoming measurements from SPT-3G will
reduce the uncertainties in our reconstructions by about a factor of two.
