We present a method to estimate the relative position and orientation of two non-coplanar microphone arrays using a room impulse response (RIR) from a single source. The approach exploits early acoustic reflections, retrieved by each array in the form of a 3D image-source point cloud, using a recently-proposed image-source localization technique. The task is then reduced to aligning the two unlabeled point clouds, which is made difficult by their inherent symmetries and by spurious, missing and noisy estimates. To this aim we introduce ARCTIC, a point-cloud registration algorithm that performs a ranked search over congruent triangles anchored on the true source. Simulated and real experiments show that ARCTIC strongly outperforms two baselines, achieving mean angular and radial self-localization errors below 5 and 10 cm across 36 real RIRs measured in 6 different rooms.