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In this paper, we provide a principled and unified explanation how knowledge in global 3-D structural invariants, typically captured by a group action on a symmetric structure, can significantly facilitate the task of reconstructing a 3-D scene from one or more images. More importantly, the "absolute" pose between the camera frame and the canonical frame that symmetric objects (e.g., buildings, hallways) provide us overwhelming clues to their orientation and position. We give the necessary and sufficient conditions under which this pose can be uniquely determined, and when such conditions are not satisfied, exactly to what extent this pose can be recovered. We show how algorithms from conventional multiple-view geometry, after properly modified and extended, can be effectively applied to perform such recovery. Since the structure, pose and even camera calibration can be recovered from a single image; the techniques naturally apply to vision-based robot navigation where global position and orientation is important.