Single-source domain generalization (SDG) aims to learn a model from a single source domain that can generalize well on unseen target domains. This is an important task in computer vision, particularly relevant to medical imaging where domain shifts are common. In this work, we consider a challenging yet practical setting: SDG for cross-modality medical image segmentation. We combine causality-inspired theoretical insights on learning domain-invariant representations with recent advancements in diffusion-based augmentation to improve generalization across diverse imaging modalities. Guided by the ``intervention-augmentation equivariant'' principle, we use controlled diffusion models (DMs) to simulate diverse imaging styles while preserving the content, leveraging rich generative priors in large-scale pretrained DMs to comprehensively perturb the multidimensional style variable. Extensive experiments on challenging cross-modality segmentation tasks demonstrate that our approach consistently outperforms state-of-the-art SDG methods across three distinct anatomies and imaging modalities. The source code is available at \href{https://github.com/ratschlab/ICMSeg}{https://github.com/ratschlab/ICMSeg}.