Continual self-supervised learning (CSSL) in medical imaging trains a foundation model sequentially, alleviating the need for collecting multi-modal images for joint training and offering promising improvements in downstream performance while preserving data privacy. However, most existing methods still rely on replaying data from previous stages to prevent catastrophic forgetting, which compromises privacy and limits their applicability in real-world scenarios where data transfer across sites is often restricted. In this work, we propose InvCoSS, an inversion-driven continual self-supervised learning framework for medical multi-modal image pre-training. Specifically, after training on a previous task, InvCoSS inverts the pre-trained self-supervised model to generate synthetic images that approximate the original training distribution. These synthetic images are then combined with data from the new task for joint optimization, which effectively mitigates catastrophic forgetting while strictly adhering to the constraint of no access to previous real data. Furthermore, to improve the fidelity of synthetic images, we introduce a novel InvUNet with a multi-scale fusion architecture to restore both high- and low-frequency components of the inverted images. To enhance diversity and prevent mode collapse, we design a repulsive representation-learning mechanism that encourages a diverse feature space for synthetic images without class guidance. Extensive experiments across nine downstream tasks validate the effectiveness of InvCoSS, achieving performance comparable to or even superior to prior data-replay methods while significantly reducing storage requirements and eliminating data privacy constraints.