Omnidirectional or 360 cameras provide embodied AI agents with a holistic, wide field-of-view (FoV) view of their surroundings, motivating the use of Multi-modal Large Language Models (MLLMs) for omnidirectional spatial reasoning. However, most MLLMs are trained on conventional 2D perspective images and struggle with the severe distortions and wrap-around discontinuities induced by spherical geometry. Enabling them to generalize to non-Euclidean 3D spaces without retraining therefore remains challenging. We propose SphMind, a training-free, plug-and-play framework that decouples semantic perception from geometric reasoning. Rather than requiring MLLMs to learn spherical geometry internally, SphMind preserves their semantic capabilities while handling geometry externally. We introduce a Spherical Harmonics-based Spatial Graph (SHSG) that models spatial relationships through equivariant transformations on the sphere, together with Inference-Time Geometric Grounding (IGG), a model-agnostic closed-loop optimization process that aligns MLLM representations with spherical geometric constraints during inference. Experiments on three benchmarks show that SphMind achieves over 21.4% average improvement in directional reasoning on MP3D and Stanford2D-3D, outperforms prompt-engineering baselines by 8.7% on the real-world ODI-Bench, and improves rotational invariance by 5.9% under panorama rotations, without additional training or dataset-specific tuning. In-the-wild evaluations further show that SphMind resolves directional reasoning queries that baseline vision-language models fail to answer correctly.

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