Hand-object interaction (HOI) reconstruction and synthesis are becoming central to embodied AI and AR/VR. Yet, despite rapid progress, existing HOI generation research remains fragmented across three disjoint tracks: (1) pose-only synthesis that predicts MANO trajectories without producing pixels; (2) single-image HOI generation that hallucinates appearance from masks or 2D cues but lacks dynamics; and (3) video generation methods that require both the entire pose sequence and the ground-truth first frame as inputs, preventing true sim-to-real deployment. Inspired by the philosophy of Joo et al. (2018), we think that HOI generation requires a unified engine that brings together pose, appearance, and motion within one coherent framework. Thus we introduce PAM: a Pose-Appearance-Motion Engine for controllable HOI video generation. The performance of our engine is validated by: (1) On DexYCB, we obtain an FVD of 29.13 (vs. 38.83 for InterDyn), and MPJPE of 19.37 mm (vs. 30.05 mm for CosHand), while generating higher-resolution 480x720 videos compared to 256x256 and 256x384 baselines. (2) On OAKINK2, our full multi-condition model improves FVD from 68.76 to 46.31. (3) An ablation over input conditions on DexYCB shows that combining depth, segmentation, and keypoints consistently yields the best results. (4) For a downstream hand pose estimation task using SimpleHand, augmenting training with 3,400 synthetic videos (207k frames) allows a model trained on only 50% of the real data plus our synthetic data to match the 100% real baseline.