A class of twisting Einstein vacuum metrics, with Weyl tensors of Petrov type II or III, is examined. This class includes the only known exact twisting type III vacuum metrics, which have independently been given by Held (1974) and Robinson (1975). It is shown that a number of the metrics in this class each admit a two-parameter group of homothetic motions. Since each type III metric of this class is determined by a function which satisfies a linear differential equation, the linear addition of two such functions, each of which determines a metric which admits a homothetic motion, then determines a metric which, in general, does not admit a homothetic motion. The relationship of general type II and III metrics of this class to metrics with homothetic symmetries is also discussed.