Measurements of differential cross sections for inclusive very forward jet production in proton-lead collisions as a function of jet energy are presented. The data were collected with the CMS experiment at the LHC in the laboratory pseudorapidity range − 6.6 < η < −5.2. Asymmetric beam energies of 4 TeV for protons and 1.58 TeV per nucleon for Pb nuclei were used, corresponding to a center-of-mass energy per nucleon pair of √ s NN = 5.02 TeV. Collisions with either the proton (p+Pb) or the ion (Pb+p) traveling towards the negative η hemisphere are studied. The jet cross sections are unfolded to stable-particle level cross sections with p T 3 GeV, and compared to predictions from various Monte Carlo event generators. In addition, the cross section ratio of p+Pb and Pb+p data is presented. The results are discussed in terms of the saturation of gluon densities at low fractional parton momenta. None of the models under consideration describes all the data over the full jet-energy range and for all beam configurations. Discrepancies between the differential cross sections in data and model predictions of more than two orders of magnitude are observed.The central feature of the CMS apparatus is a superconducting solenoid of 6 m internal diameter, providing a magnetic field of 3.8 T. Within the solenoid volume are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and scintillator hadron calorimeter (HCAL), each composed of a barrel and two endcap sections. Forward calorimeters extend the η coverage provided by the barrel and endcap detectors. Muons are detected in gas-ionization chambers embedded in the steel flux-return yoke outside the solenoid.The forward hadron (HF) calorimeter uses steel as an absorber and quartz fibers as the sensitive material. The two HF detectors are located at 11.2 m from the interaction point, one on each side of CMS, and together they provide coverage in the range 3.0 < |η| < 5.2.The very forward angles, i.e., −6.6 < η < −5.2, are covered at one end of CMS by the CAS-