As the demand for 3D maps from LIDAR scanners increases, delivering high-quality maps becomes critical. One way to ensure the quality of such maps is through point cloud alignment classification, which aims to classify the alignment error between two registered point clouds. Specifically, we present the classifier FACT (Feature-Aware Classification Transformer), consisting of two main modules: feature extraction and classification. Descriptive features are extracted from the joint point cloud, which are then processed by a point transformer-based neural network to predict the alignment error class. In a ten-class point cloud alignment classification test, FACT achieved 92.4% accuracy, where the alignment error ranged from zero meters and radians to 0.9 meters and 0.09 radians. Remarkably, the classifier only made one misprediction beyond neighboring classes, exhibiting its ability to detect alignment errors as the classes have an inherent order. Furthermore, when benchmarked on two binary classification tasks, FACT showed significantly superior performance over the baseline and even obtained 100.0% accuracy for the easier of the two tasks. FACT not only detects potential errors in 3D maps but also estimates their magnitude, leading to more reliable 3D maps with quality estimations for each transformation.