As one of the fundamental components of object detection, intersection-over-union (IoU) calculations between two bounding boxes play an important role in samples selection, NMS operation and evaluation of object detection algorithms. This procedure is well-defined and solved for planar images, while it is challenging for spherical ones. Some existing methods utilize planar bounding boxes to represent spherical objects. However, they are biased due to the distortions of spherical objects. Others use spherical rectangles as unbiased representations, but they adopt excessive approximate algorithms when computing the IoU. In this paper, we propose an unbiased IoU as a novel evaluation criterion for spherical image object detection, which is based on the unbiased representations and utilize unbiased analytical method for IoU calculation. This is the first time that the absolutely accurate IoU calculation is applied to the evaluation criterion, thus object detection algorithms can be correctly evaluated for spherical images. With the unbiased representation and calculation, we also present Spherical CenterNet, an anchor free object detection algorithm for spherical images. The experiments show that our unbiased IoU gives accurate results and the proposed Spherical CenterNet achieves better performance on one real-world and two synthetic spherical object detection datasets than existing methods.