Recent advances in imitation learning and vision-language models highlight the need for high-fidelity tactile perception, with 6-DoF tactile object pose estimation providing a crucial foundation for precise robotic manipulation. We introduce InvariantCloud, a 6-DoF pose estimation framework that leverages the global invariance of surface marker constellations on vision-based tactile sensors. In contrast to recent approaches, our one-shot globally invariant point cloud registration suppresses cumulative drift and overcomes long-standing limitations in accurately estimating yaw (Z-axis) rotation. Experimental verifications show that InvariantCloud achieves superior yaw tracking accuracy and re-localization repeatability compared to existing benchmarks, demonstrating its precision and robustness in long-sequence manipulation tasks.