With the advance of omnidirectional panoramic technology, 360◦ imagery has become increasingly popular in the past few years. To better understand the 360◦ content, many works resort to the 360◦ object detection and various criteria have been proposed to bound the objects and compute the intersection-over-union (IoU) between bounding boxes based on the common equirectangular projection (ERP) or perspective projection (PSP). However, the existing 360◦ criteria are either inaccurate or inefficient for real-world scenarios. In this paper, we introduce a novel spherical criteria for fast and accurate 360◦ object detection, including both spherical bounding boxes and spherical IoU (SphIoU). Based on the spherical criteria, we propose a novel two-stage 360◦ detector, ie, Reprojection R-CNN, by combining the advantages of both ERP and PSP, yielding efficient and accurate 360◦ object detection. To validate the design of spherical criteria and Reprojection R-CNN, we construct two unbiased synthetic datasets for training and evaluation. Experimental results reveal that compared with the existing criteria, the two-stage detector with spherical criteria achieves the best mAP results under the same inference speed, demonstrating that the spherical criteria can be more suitable for 360◦ object detection. Moreover, Reprojection R-CNN outperforms the previous state-of-the-art methods by over 30% on mAP with competitive speed, which confirms the efficiency and accuracy of the design.