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 (or high-l) CMB datasets from either WMAP or Planck as well as CMB lensing as traced by Planck's measured four point correlation function. 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, roughly of order 2 sigma, robust to the choice of the complementary high resolution 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(-0.013) (+0.012) , again very stable against the particular choice for high-l 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 1 sigma. In particular, cleaning dust with Planck's 353 GHz data imposes a 1 sigma downward shift in the value of the Hubble constant H-0, significantly contributing to the tension reported between CMB based and direct measurements of the present expansion rate. On the other hand, we find that the appearance of the so-called low l 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-l dataset employed.