The Hamamatsu R11410-21 photomultiplier tube is the photodetector of choice for the XENON1T dual-phase time projection chamber. The device has been optimized for a very low intrinsic radioactivity, a high quantum efficiency and a high sensitivity to single photon detection. A total of 248 tubes are currently operated in XENON1T, selected out of 321 tested units. In this article the procedures implemented to evaluate the large number of tubes prior to their installation in XENON1T are described. The parameter distributions for all tested tubes are shown, with an emphasis on those selected for XENON1T, of which the impact on the detector performance is discussed. All photomultipliers have been tested in a nitrogen atmosphere at cryogenic temperatures, with a subset of the tubes being tested in gaseous and liquid xenon, simulating their operating conditions in the dark matter detector. The performance and evaluation of the tubes in the different environments is reported and the criteria for rejection of PMTs are outlined and quantified.