  Cosmological parameter measurements from CMB experiments such as Planck,
ACTpol, SPTpol and other high resolution follow-ons fundamentally rely on the
accuracy of the assumed recombination model, or one with well prescribed
uncertainties. Deviations from the standard recombination history might suggest
new particle physics or modified atomic physics. Here we treat possible
perturbative fluctuations in the free electron fraction, $\Xe(z)$, by a
semi-blind expansion in densely-packed modes in redshift. From these we
construct parameter eigenmodes, which we rank order so that the lowest modes
provide the most power to probe the $\Xe(z)$ with CMB measurements. Since the
eigenmodes are effectively weighed by the fiducial $\Xe$ history, they are
localized around the differential visibility peak, allowing for an excellent
probe of hydrogen recombination, but a weaker probe of the higher redshift
helium recombination and the lower redshift highly neutral freeze-out tail. We
use an information-based criterion to truncate the mode hierarchy, and show
that with even a few modes the method goes a long way towards morphing a
fiducial older {\sc Recfast} $X_{\rm e,i} (z)$ into the new and improved {\sc
CosmoRec} and {\sc HyRec} $X_{\rm e,f} (z)$ in the hydrogen recombination
regime, though not well in the helium regime. Without such a correction, the
derived cosmic parameters are biased. We discuss an iterative approach for
updating the eigenmodes to further hone in on $X_{\rm e,f} (z)$ if large
deviations are indeed found. We also introduce control parameters that
downweight the attention on the visibility peak structure, e.g., focusing the
eigenmode probes more strongly on the $\Xe (z)$ freeze-out tail, as would be
appropriate when looking for the $\Xe$ signature of annihilating or decaying
elementary particles.
