A numerical study of an ensemble of orbits based on observed objects in the near-Neptune high-eccentricity (NNHE) region, with perihelion distances q in the range 28 < q < 35.5 au and semimajor axes a in the range 60 < a < 1000 an, is used to predict the orbital distribution of Centaurs (5 < q < 28 au) for comparison with observations after correcting for discovery biases. The majority of Centaurs produced in this way have a less than or similar to 60 au. However, the intrinsic number of observed Centaurs is dominated by longer period objects, the number with a > 60 au being roughly an order of magnitude greater than that for a < 60 au, and therefore inconsistent with a source in the NNHE region, which is broadly similar to the so-called 'Scattered Disc'. The observed distribution of Centaurs with a less than or similar to 60 au is also inconsistent with this source, although it is conceivable that in this region the discrepancies might be explained by factors such as out-gassing, splitting or varying albedo not included in our model. Thus, although Centaurs can be produced from the NNHE region, their numbers and orbital distributions are inconsistent with this region being the dominant source for all Centaurs. We conclude that there must be another source flux, especially for the longer period, more populous group, and suggest that the most likely source for these objects is the Oort cloud. Thus, there are two separate, but overlapping, dynamical classes of Centaurs, one originating from the Oort cloud and the other from the NNHE region. The two source regions produce roughly similar contributions to Centaurs with a less than or similar to 60 an and to the observed Jupiter family of comets.