Distribution-to-Distribution (D2D) point cloud registration algorithms are fast, interpretable, and perform well in unstructured environments. Unfortunately, existing strategies for predicting solution error for these methods are overly optimistic, particularly in regions containing large or extended physical objects. In this paper we introduce the Iterative Closest Ellipsoidal Transform (ICET), a novel 3D LIDAR scan-matching algorithm that re-envisions NDT in order to provide robust accuracy prediction from first principles. Like NDT, ICET subdivides a LIDAR scan into voxels in order to analyze complex scenes by considering many smaller local point distributions, however, ICET assesses the voxel distribution to distinguish random noise from deterministic structure. ICET then uses a weighted least-squares formulation to incorporate this noise/structure distinction into computing a localization solution and predicting the solution-error covariance. In order to demonstrate the reasonableness of our accuracy predictions, we verify 3D ICET in three LIDAR tests involving real-world automotive data, high-fidelity simulated trajectories, and simulated corner-case scenes. For each test, ICET consistently performs scan matching with sub-centimeter accuracy. This level of accuracy, combined with the fact that the algorithm is fully interpretable, make it well suited for safety-critical transportation applications. Code is available at https://github.com/mcdermatt/ICET