Humans universally dislike the task of cleaning up a messy room. If machines were to help us with this task, they must understand human criteria for regular arrangements, such as several types of symmetry, co-linearity or co-circularity, spacing uniformity in linear or circular patterns, and further inter-object relationships that relate to style and functionality. Previous approaches for this task relied on human input to explicitly specify goal state, or synthesized scenes from scratch--but such methods do not address the rearrangement of existing messy scenes without providing a goal state. In this paper, we present LEGO-Net, a data-driven transformer-based iterative method for LEarning reGular rearrangement of Objects in messy rooms. LEGO-Net is partly inspired by diffusion models--it starts with an initial messy state and iteratively “de-noises” the position and orientation of objects to a regular state while reducing distance traveled. Given randomly perturbed object positions and orientations in an existing dataset of professionally-arranged scenes, our method is trained to recover a regular re-arrangement. Results demonstrate that our method is able to reliably rearrange room scenes and outperform other methods. We additionally propose a metric for evaluating regularity in room arrangements using number-theoretic machinery.