Accurate attitude estimation using low-cost sensors is an important capability to enable many robotic applications. In this paper, we present a method based on the concept of corren-tropy in Kalman filtering to estimate the 3D orientation of a rigid body using a low-cost inertial measurement unit (IMU). We then leverage the proposed attitude estimation framework to develop a LiDAR-Intertial Odometry (LIO) demonstrating improved lo-calization accuracy with respect to traditional methods. This is of particular importance when the robot undergoes high-rate motions that typically exacerbate the issues associated with low-cost sensors. The proposed orientation estimation approach is first validated using the data coming from a low-cost IMU sensor. We further demonstrate the performance of the proposed LIO solution in a simulated robotic cave exploration scenario.