Humans learn object orientation progressively, from recognizing which way an object faces, to mentally rotating it, to reasoning about orientations between objects. Current vision-language benchmarks largely conflate orientation with position and general scene understanding. We introduce Discriminative Orientation Reasoning Intelligence (DORI), a cognitively grounded hierarchical benchmark that makes object orientation the primary target. Inspired by stages of human orientation cognition, DORI decomposes orientation into four dimensions, each evaluated at coarse (categorical) and granular (metric) levels. Composed from 13,652 images across 14 sources, DORI provides 33,656 multiple-choice questions covering 67 object categories in real-world and synthetic settings. Its coarse-to-granular design isolates orientation from confounds such as object recognition difficulty, scene clutter, and linguistic ambiguity via bounding-box isolation, standardized spatial reference frames, and structured prompts. Evaluating 24 state-of-the-art vision-language models shows a clear pattern: models that perform well on general spatial benchmarks are near-random on object-centric orientation tasks. The best models reach only 54.2% on coarse and 45.0% on granular judgments, with largest failures on compound rotations and shifts in inter-object reference frames. Large coarse-to-granular gaps reveal reliance on categorical heuristics rather than geometric reasoning, a limitation hidden by existing benchmarks. These results identify orientation understanding as an unsolved challenge for multimodal systems, with implications for robotic manipulation, 3D scene reconstruction, and human-AI interaction.