The scarcity of labeled data often limits the application of supervised deep learning techniques for medical image segmentation. This has motivated the development of semi-supervised techniques that learn from a mixture of labeled and unlabeled images. In this paper, we propose a novel semi-supervised method that, in addition to supervised learning on labeled training images, learns to predict segmentations consistent under a given class of transformations on both labeled and unlabeled images. More specifically, in this work we explore learning equivariance to elastic deformations. We implement this through: (1) a Siamese architecture with two identical branches, each of which receives a differently transformed image, and (2) a composite loss function with a supervised segmentation loss term and an unsupervised term that encourages segmentation consistency between the predictions of the two branches. We evaluate the method on a public dataset of chest radiographs with segmentations of anatomical structures using 5-fold cross-validation. The proposed method reaches significantly higher segmentation accuracy compared to supervised learning. This is due to learning transformation consistency on both labeled and unlabeled images, with the latter contributing the most. We achieve the performance comparable to state-of-the-art chest X-ray segmentation methods while using substantially fewer labeled images.