Self-supervised learning (SSL) and diffusion models have advanced representation learning and image synthesis, but in 3D medical imaging they are still largely used separately for analysis and synthesis, respectively. Unifying them is appealing but difficult, because multi-source data exhibit pronounced style shifts while downstream tasks rely primarily on anatomy, causing anatomical content and acquisition style to become entangled. In this paper, we propose MeDUET, a 3D Medical image Disentangled UnifiEd PreTraining framework in the variational autoencoder latent space. Our central idea is to treat unified pretraining under heterogeneous multi-center data as a factor identifiability problem, where content should consistently capture anatomy and style should consistently capture appearance. MeDUET addresses this problem through three components. Token demixing provides controllable supervision for factor separation, Mixed Factor Token Distillation reduces factor leakage under mixed regions, and Swap-invariance Quadruplet Contrast promotes factor-wise invariance and discriminability. With these learned factors, MeDUET transfers effectively to both synthesis and analysis, yielding higher fidelity, faster convergence, and better controllability for synthesis, while achieving competitive or superior domain generalization and label efficiency on diverse medical benchmarks. Overall, MeDUET shows that multi-source heterogeneity can serve as useful supervision, with disentanglement providing an effective interface for unifying 3D medical image synthesis and analysis. Our code is available at https://github.com/JK-Liu7/MeDUET.

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