The almost hermetic coverage of the CMS detector is used to measure the distribution of transverse energy, ET, over 13.2 units of pseudorapidity, η, for pPb collisions at a center-of-mass energy per nucleon pair of √sNN=5.02 TeV. The huge angular acceptance exploits the fact that the CASTOR calorimeter at -6.6 < η < -5.2 is effectively present on both sides of the colliding system because of a switch in the proton-going and lead-going beam directions. This wide acceptance enables the study of correlations between well-separated angular regions and makes the measurement a particularly powerful test of event generators. For minimum bias pPb collisions the maximum value of dET/dη is 22 GeV, which implies an ET per participant nucleon pair comparable to that of peripheral PbPb collisions at √sNN= 2.76 TeV. The increase of dET/dη with centrality is much stronger for the lead-going side than for the proton-going side. The η dependence of dET/dη is sensitive to the η range in which the centrality variable is defined. Several modern generators are compared to these results but none is able to capture all aspects of the η and centrality dependence of the data and the correlations observed between different η regions.