Radiological analysis increasingly benefits from pretrained visual representations that can support heterogeneous downstream tasks across imaging modalities. In this work, we introduce OmniRad, a self-supervised radiological foundation model pretrained on 1.2 million medical images, designed with radiology-inspired principles emphasizing representation reuse and cross-task transferability. We evaluate the pretrained encoder under multiple downstream adaptation regimes, including lightweight task-specific adapters with a frozen backbone as well as full end-to-end fine-tuning for classification, allowing us to assess both representation quality and task-specific performance. OmniRad is evaluated on a broad suite of public benchmarks spanning classification and segmentation across multiple modalities. On the MedMNISTv2 collection, OmniRad improves classification F1 by up to 2.05% over competing foundation models. For dense prediction, OmniRad attains mean Dice score improvements across six MedSegBench datasets when using frozen representations. Qualitative analyses and latent-space visualizations suggest improved feature clustering and modality-related separation.