We consider path planning for a rigid spatial robot moving amidst polyhedral obstacles. Our robot is either a rod or a ring. Being axially-symmetric, their configuration space is ℝ3 ×S2 with 5 degrees of freedom (DOF). Correct, complete and practical path planning for such robots is a long standing challenge in robotics. While the rod is one of the most widely studied spatial robots in path planning, the ring seems to be new, and a rare example of a non-simply-connected robot. This work provides rigorous and complete algorithms for these robots with theoretical guarantees. We implemented the algorithms in our open-source Core Library. Experiments show that they are practical, achieving near real-time performance. We compared our planner to state-of-the-art sampling planners in OMPL . Our subdivision path planner is based on the twin foundations of ε-exactness and soft predicates. Correct implementation is relatively easy. The technical innovations include subdivision atlases for S2, introduction of Σ2 representations for footprints, and extensions of our feature-based technique for “opening up the blackbox of collision detection”.