  Recent measurements from the CMB and from high-redshift galaxy observations
have placed rough constraints on the midpoint and duration of the Epoch of
Reionization. Detailed measurements of the ionization history remain elusive,
although two proposed probes show great promise for this purpose: the 21cm
global signal and the kinetic Sunyaev-Zel'dovich (kSZ) effect. We formally
confirm the common assumption that these two probes are highly complementary,
with the kSZ being more sensitive to extended ionization histories and the
global signal to rapidly evolving ones. We do so by performing a
Karhunen-Lo\`{e}ve (KL) transformation, which casts the data in a basis
designed to emphasize the information content of each probe. We find that
reconstructing the ionization history using both probes gives significantly
more precise results than individual constraints, although carefully chosen,
physically motivated priors play a crucial part in obtaining a bias-free
reconstruction. Additionally, in the KL basis, measurements from one probe can
be used to detect the presence of residual systematics in the other, providing
a safeguard against systematics that would go undetected when data from each
probe is analyzed in isolation. Once detected, the modes contaminated by
systematics can be discarded from the data analysis to avoid biases in
reconstruction.
