We present COIN-LIO, a LiDAR Inertial Odometry pipeline that tightly couples information from LiDAR intensity with geometry-based point cloud registration. The focus of our work is to improve the robustness of LiDAR-inertial odometry in geometrically degenerate scenarios, like tunnels or flat fields. We project LiDAR intensity returns into an intensity image, and propose an image processing pipeline that produces filtered images with improved brightness consistency within the image as well as across different scenes. To effectively leverage intensity as an additional modality, we present a novel feature selection scheme that detects uninformative directions in the point cloud registration and explicitly selects patches with complementary image information. Photometric error minimization in the image patches is then fused with inertial measurements and point-to-plane registration in an iterated Extended Kalman Filter. The proposed approach improves accuracy and robustness on a public dataset. We additionally publish a new dataset, that captures five real-world environments in challenging, geometrically degenerate scenes. By using the additional photometric information, our approach shows drastically improved robustness against geometric degeneracy in environments where all compared baseline approaches fail.