Gravitational lensing of the cosmic microwave background (CMB) has recently started to gain importance as a cosmological probe. With growing detection significance of this effect, it is necessary to further develop theoretical understanding of its consequences. Such studies are the main topic of this work, that is based on the papers [1, 2, 3].,We start by looking at correlations that the gravitational lensing induces between CMB temperature, polarization and reconstructed lensing potential and investigate how neglecting them in an analysis impacts constraints on cosmological parameters. We find that for the planned CMB Stage 4 experiment, neglecting these correlations can significantly underestimate variance of certain combinations of cosmological parameters, as well as lead to an increased frequency of mistakenly rejecting the underlying cosmological model.,Then we discuss a method we developed to directly measure the gravitational lensing potential from the CMB data and explain how to practically perform such measurement. This method helps us understand why it is necessary to include the lensing-induced covariances to get correct constraints on cosmological parameters. Additionally, comparing direct measurements of the lensing potential from various subsets of data or across experiments allows for powerful consistency checks that can be used to search for residual systematics and exotic new physics. When assuming a particular cosmological model, this technique can also be used to probe internal consistency of lensing within a single data set.,In the final part of this work, we apply this methodology to check lensing consistency of the Planck satellite data. We find that it is not possible to resolve the lensing anomalies seen in this data even when allowing for an arbitrary gravitational lensing potential, beyond the predictions of the standard cosmological model. Significances of these tensions are evaluated at above 2σ; one possible explanation are residual systematics in the Planck temperature power spectrum. Without large modifications, this technique can be applied to data from other current and especially future experiments, where its full power will become manifest.