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Permutation ambiguity is a major challenge in training monaural talker-independent speaker separation. While permutation invariant training (PIT) is a widely used technique, it functions as a `black box', providing little insight into which auditory cues lead to successful training. We introduce a new approach to speaker separation by leveraging differences in pitch and onset, which are both prominent cues for auditory scene analysis. We propose pitch-based and onset-based training to resolve permutation ambiguity, assigning speakers by their pitch frequencies and onset times, respectively. This approach offers a more explainable training strategy than PIT. We also propose a hybrid criterion combining these cues to improve separation performance in challenging conditions such as same-gender speakers or close utterance onsets. Evaluation results show that pitch and onset criteria each perform competitively to PIT and the hybrid criterion surpasses PIT in separating two-speaker mixtures.