  We present forecasts for cosmological parameters from future Cosmic Microwave
Background (CMB) data measured by the Stage-4 (S4) generation of ground-based
experiments in combination with large-scale anisotropy data from the PIXIE
satellite. We demonstrate the complementarity of the two experiments and focus
on science targets that benefit from their combination. We show that a
cosmic-variance-limited measurement of the optical depth to reionization
provided by PIXIE, with error $\sigma(\tau)=0.002$, is vital for enabling a
5$\sigma$ detection of the sum of the neutrino masses when combined with a
CMB-S4 lensing measurement, and with lower-redshift constraints on the growth
of structure and the distance-redshift relation. Parameters characterizing the
epoch of reionization will also be tightly constrained; PIXIE's $\tau$
constraint converts into $\sigma(\rm{z_{re}})=0.2$ for the mean time of
reionization, and a kinematic Sunyaev-Zel'dovich measurement from S4 gives
$\sigma(\Delta \rm{z_{re}})=0.03$ for the duration of reionization. Both PIXIE
and S4 will put strong constraints on primordial tensor fluctuations, vital for
testing early-universe models, and will do so at distinct angular scales. We
forecast $\sigma(r)\approx 5\times10^{-4}$ for a signal with a tensor-to-scalar
ratio $r=10^{-3}$, after accounting for diffuse foreground removal and
de-lensing. The wide and dense frequency coverage of PIXIE results in an
expected foreground-degradation factor on $r$ of only $\approx$25%. By
measuring large and small scales PIXIE and S4 will together better limit the
energy injection at recombination from dark matter annihilation, with $p_{\rm
ann}<0.09 \times 10^{-6} \ {\rm m^3/s/Kg}$ projected at 95% confidence.
Cosmological parameters measured from the damping tail with S4 will be best
constrained by polarization, which has the advantage of minimal contamination
from extragalactic emission.
