Unsupervised visible-infrared person re-identification (UVI-ReID) aims to retrieve pedestrian images across different modalities without costly annotations, but faces challenges due to the modality gap and lack of supervision. Existing methods often adopt self-training with clustering-generated pseudo-labels but implicitly assume these labels are always correct. In practice, however, this assumption fails due to inevitable pseudo-label noise, which hinders model learning. To address this, we introduce a new learning paradigm that explicitly considers Pseudo-Label Noise (PLN), characterized by three key challenges: noise overfitting, error accumulation, and noisy cluster correspondence. To this end, we propose a novel Robust Duality Learning framework (RoDE) for UVI-ReID to mitigate the effects of noisy pseudo-labels. First, to combat noise overfitting, a Robust Adaptive Learning mechanism (RAL) is proposed to dynamically emphasize clean samples while down-weighting noisy ones. Second, to alleviate error accumulation-where the model reinforces its own mistakes-RoDE employs dual distinct models that are alternately trained using pseudo-labels from each other, encouraging diversity and preventing collapse. However, this dual-model strategy introduces misalignment between clusters across models and modalities, creating noisy cluster correspondence. To resolve this, we introduce Cluster Consistency Matching (CCM), which aligns clusters across models and modalities by measuring cross-cluster similarity. Extensive experiments on three benchmarks demonstrate the effectiveness of RoDE.