Large pretrained foundation models demonstrate exceptional performance and, in some high-stakes applications, even surpass human experts. However, most of these models are currently evaluated primarily on prediction accuracy, overlooking the validity of the rationales behind their accurate predictions. For the safe deployment of foundation models, there is a pressing need to ensure double-correct predictions, i.e., correct prediction backed by correct rationales. To achieve this, we propose a two-phase scheme: First, we curate a new dataset that offers structured rationales for visual recognition tasks. Second, we propose a rationale-informed optimization method to guide the model in disentangling and localizing visual evidence for each rationale, without requiring manual annotations. Extensive experiments and ablation studies demonstrate that our model outperforms state-of-the-art models by up to 10.1% in prediction accuracy across a wide range of tasks. Furthermore, our method significantly improves the model's rationale correctness, improving localization by 7.5% and disentanglement by 36.5%. Our dataset, source code, and pretrained weights: https://github.com/deep-real/DCP