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Breast cancer remains a leading cause of death among women, with early detection significantly improving prognosis. Non-contrast computed tomography (NCCT) scans of the chest, routinely acquired for thoracic assessments, often capture the breast region incidentally, presenting an underexplored opportunity for opportunistic breast lesion detection without additional imaging cost or radiation. However, the subtle appearance of lesions in NCCT and the difficulty of jointly modeling lesion detection and malignancy classification pose unique challenges. In this work, we propose Dual-Res Tandem Mamba-3D (DRT-M3D), a novel multitask framework for opportunistic breast cancer analysis on NCCT scans. DRT-M3D introduces a dual-resolution architecture, which captures fine-grained spatial details for segmentation-based lesion detection and global contextual features for breast-level cancer classification. It further incorporates a tandem input mechanism that models bilateral breast regions jointly through Mamba-3D blocks, enabling cross-breast feature interaction by leveraging subtle asymmetries between the two sides. Our approach achieves state-of-the-art performance in both tasks across multi-institutional NCCT datasets spanning four medical centers. Extensive experiments and ablation studies validate the effectiveness of each key component.