Rigid registration aims to determine the translations and rotations necessary to align features in a pair of images. While recent machine learning methods have become state-of-the-art for linear and deformable registration across subjects, they have demonstrated limitations when applied to longitudinal (within-subject) registration, where achieving precise alignment is critical. Building on an existing framework for anatomy-aware, acquisition-agnostic affine registration, we propose a model optimized for longitudinal, rigid brain registration. By training the model with synthetic within-subject pairs augmented with rigid and subtle nonlinear transforms, the model estimates more accurate rigid transforms than previous cross-subject networks and performs robustly on longitudinal registration pairs within and across magnetic resonance imaging (MRI) contrasts.