Space-time transformations in Minkowski space are interpreted as neither "active" nor "passive" but as transformations of events and trajectories of processes with respect to the same observer and in the same flat space-time. The distinction is significant for the nonlinear special conformal transformations, which violate causality in the sense that the causal relations among events of a process are not always the same as among corresponding events of a transformed process. A special conformal transformation can always be found to transform the separation of any pair of events of nonlightlike separation from timelike to spacelike and vice versa. The transformation of trajectories under special conformal transformations is studied. Examples of reversal of temporal ordering by special conformal transformations are presented. Finally it is argued that none of this should prevent the conformal group from being physically useful.