We develop a neural network architecture which, trained in an unsupervised manner as a denoising diffusion model, simultaneously learns to both generate and segment images. Learning is driven entirely by the denoising diffusion objective, without any annotation or prior knowledge about regions during training. A computational bottleneck, built into the neural architecture, encourages the denoising network to partition an input into regions, denoise them in parallel, and combine the results. Our trained model generates both synthetic images and, by simple examination of its internal predicted partitions, semantic segmentations of those images. Without fine-tuning, we directly apply our unsupervised model to the downstream task of segmenting real images via noising and subsequently denoising them. Experiments demonstrate that our model achieves accurate unsupervised image segmentation and high-quality synthetic image generation across multiple datasets.