Digraph Representation Learning aims to learn representations for directed homogeneous graphs (digraphs). Prior work is largely constrained or has poor generalizability across tasks. Most Graph Neural Networks exhibit poor performance on digraphs due to the neglect of modeling neighborhoods and preserving asymmetry. In this paper, we address these notable challenges by leveraging hyperbolic collaborative learning from multi-ordered partitioned neighborhoods and asymmetry-preserving regularizers. Our resulting formalism, Digraph Hyperbolic Networks (D-HYPR), is versatile for multiple tasks including node classification, link presence prediction, and link property prediction. The efficacy of D-HYPR was meticulously examined against 21 previous techniques, using 8 real-world digraph datasets. D-HYPR statistically significantly outperforms the current state of the art. We release our code at https://github. com/hongluzhou/dhypr.