Inertial odometry (IO) using only Inertial Measurement Units (IMUs) enables camera-free head tracking in head-mounted devices such as VR headsets, AR glasses, and earbuds. Prior IO methods operate in a gravity-aligned world frame, which is not yaw-equivariant. Yaw augmentation improves robustness but does not guarantee equivariance; learned equivariant layers achieve it, but EqNIO's full pipeline costs nearly 3$\times$ the FLOPs of ours. While raw body-frame IMU inputs are inherently yaw-invariant, head tilt redistributes both gravity and motion signals across device axes, making the representation tilt-variant. We propose HeadIO, which combines a body-frame heading decomposition with a bidirectional Mamba backbone for head-mounted inertial odometry. Our decomposition achieves tilt invariance by separating gravity-aligned and horizontal components using only the tilt component of device orientation. We evaluate on four head-mounted datasets and find that HeadIO improves RTE-5s by 8.1% (0.260 m $\to$ 0.239 m) over world-frame methods with learned equivariant layers on Nymeria, the largest benchmark dataset, at nearly 3$\times$ lower FLOPs and 8$\times$ fewer parameters. HeadIO achieves state-of-the-art performance on our primary metric, RTE-5s, on all Aria-glasses datasets, which natively provide body-frame semantics for our decomposition method. We further evaluate the IMU-only deployment scenario on Nymeria, where HeadIO remains state-of-the-art. We show that our method relies mainly on tilt accuracy, which can be obtained from the gravity signal; for yaw, a single-coefficient gyroscope correction removes 11.6% of the yaw error and improves RTE-5s by 2.5%, and sparse external yaw corrections (for example, one every 30 s) can improve RTE-5s by 12.0%. Our decomposition is lightweight, composes with any backbone, and is suitable for real-time deployment on head-mounted devices.

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