Summary: The varying cortical geometry of the brain creates numerous challenges for its analysis. Recent developments have enabled learning cortical data directly across multiple brain surfaces via graph convolutions. However, current graph learning algorithms fail when brain surface data are misaligned across subjects, thereby requiring to apply a costly alignment procedure in pre-processing. Adversarial training is widely used for unsupervised domain adaptation to improve segmentation performance on target data whose distribution differs from the training source data. In this paper, we exploit this technique to learn surface data across inconsistent graph alignments. This novel approach comprises a segmentator that uses graph convolution layers to enable parcellation across brain surfaces of varying geometry, and a discriminator that predicts the alignment-domain of surfaces from their segmentation. By trying to fool the discriminator, the adversarial training learns an alignment-invariant representation which yields consistent parcellations for differently-aligned surfaces. Using manually-labeled brain surface from MindBoggle, the largest publicly available dataset of this kind, we demonstrate a 2%-13% improvement in mean Dice over a non-adversarial training strategy, for test brain surfaces with no alignment or aligned on a different reference than source examples.