We consider the directed percolation process as a prototype of systems displaying a nonequilibrium phase transition into an absorbing state. The model is in a critical state when the activation probability is adjusted at some precise value . Criticality is lost as soon as the probability to activate sites at the first attempt, , is changed. We show here that criticality can be restored by “compensating” the change in by an appropriate change of the second time activation probability in the opposite direction. At compensation, we observe that the bulk exponents of the process coincide with those of the normal directed percolation process. However, the spreading exponents are changed and take values that depend continuously on the pair . We interpret this situation by acknowledging that the model with modified initial probabilities has an infinite number of absorbing states.