  CMB and lensing reconstruction power spectra are powerful probes of
cosmology. However they are correlated, since the CMB power spectra are lensed
and the lensing reconstruction is constructed using CMB multipoles. We perform
a full analysis of the auto- and cross-covariances, including polarization
power spectra and minimum variance lensing estimators, and compare with
simulations of idealized future CMB-S4 observations. Covariances sourced by
fluctuations in the unlensed CMB and instrumental noise can largely be removed
by using a realization-dependent subtraction of lensing reconstruction noise,
leaving a relatively simple covariance model that is dominated by
lensing-induced terms and well described by a small number of principal
components. The correlations between the CMB and lensing power spectra will be
detectable at the level of $\sim 5\sigma$ for a CMB-S4 mission, and neglecting
those could underestimate some parameter error bars by several tens of percent.
However we found that the inclusion of external priors or data sets to estimate
parameter error bars can make the impact of the correlations almost negligible.
