(abridged) We study the impact of the large-angle CMB polarization datasets publicly released by the WMAP and Planck satellites on the estimation of cosmological parameters of the $Lambda$CDM model. To complement large-angle polarization, we consider the high-resolution CMB datasets from either WMAP or Planck, as well as CMB lensing as traced by Planck. In the case of WMAP, we compute the large-angle polarization likelihood starting over from low-resolution frequency maps and their covariance matrices, and perform our own foreground mitigation technique, which includes as a possible alternative Planck 353 GHz data to trace polarized dust. We find that the latter choice induces a downward shift in the optical depth $tau$, of order ~$2sigma$, robust to the choice of the complementary high-l dataset. When the Planck 353 GHz is consistently used to minimize polarized dust emission, WMAP and Planck 70 GHz large-angle polarization data are in remarkable agreement: by combining them we find $tau = 0.066 ^{+0.012}_{-0.013}$, again very stable against the particular choice for high-$ell$ data. We find that the amplitude of primordial fluctuations $A_s$, notoriously degenerate with $tau$, is the parameter second most affected by the assumptions on polarized dust removal, but the other parameters are also affected, typically between $0.5$ and $1sigma$. In particular, cleaning dust with plancks 353 GHz data imposes a $1sigma$ downward shift in the value of the Hubble constant $H_0$, significantly contributing to the tension reported between CMB based and direct measurements of $H_0$. On the other hand, we find that the appearance of the so-called low $ell$ anomaly, a well-known tension between the high- and low-resolution CMB anisotropy amplitude, is not significantly affected by the details of large-angle polarization, or by the particular high-$ell$ dataset employed.
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