The almost hermetic coverage of the CMS detector is used to measure the distribution of transverse energy, E-T, over 13.2 units of pseudorapidity, eta, for pPb collisions at a center-of-mass energy per nucleon pair of root s(NN) = 5.02 TeV. The huge angular acceptance exploits the fact that the CASTOR calorimeter at -6.6 < eta < -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 dE(T)/d eta is 22 GeV, which implies an E-T per participant nucleon pair comparable to that of peripheral PbPb collisions at root s(NN) 7 = 2.76 TeV. The increase of dE(T)/d eta with centrality is much stronger for the lead-going side than for the proton-going side. The i dependence of dE(T)/d eta is sensitive to the eta 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 eta and centrality dependence of the data and the correlations observed between different eta regions.