Optical flow estimation in omnidirectional videos faces two significant issues: the lack of benchmark datasets and the challenge of adapting perspective video-based methods to accommodate the omnidirectional nature. This paper proposes the first perceptually natural-synthetic omnidirectional benchmark dataset with a 360° field of view, FLOW360, with 40 different videos and 4,000 video frames. We conduct comprehensive characteristic analysis and comparisons between our dataset and existing optical flow datasets, which manifest perceptual realism, uniqueness, and diversity. To accommodate the omnidirectional nature, we present a novel Siamese representation Learning framework for Omnidirectional Flow (SLOF). We train our network in a contrastive manner with a hybrid loss function that combines contrastive loss and optical flow loss. Extensive experiments verify the proposed framework's effectiveness and show up to 40% performance improvement over the state-of-the-art approaches. We will publish the FLOW360 dataset with all raw Blender scenes and Blender add-ons for researchers to create custom optical flow datasets for perspective and omnidirectional videos.