Hashing-based techniques have provided attractive solutions to cross-modal similarity search when addressing vast quantities of multimedia data. However, existing cross-modal hashing (CMH) methods face two critical limitations: 1) there is no previous work that simultaneously exploits the consistent or modality-specific information of multi-modal data; 2) the discriminative capabilities of pairwise similarity is usually neglected due to the computational cost and storage overhead. Moreover, to tackle the discrete constraints, relaxation-based strategy is typically adopted to relax the discrete problem to the continuous one, which severely suffers from large quantization errors and leads to sub-optimal solutions. To overcome the above limitations, in this article, we present a novel supervised CMH method, namely Asymmetric Supervised Consistent and Specific Hashing (ASCSH). Specifically, we explicitly decompose the mapping matrices into the consistent and modality-specific ones to sufficiently exploit the intrinsic correlation between different modalities. Meanwhile, a novel discrete asymmetric framework is proposed to fully explore the supervised information, in which the pairwise similarity and semantic labels are jointly formulated to guide the hash code learning process. Unlike existing asymmetric methods, the discrete asymmetric structure developed is capable of solving the binary constraint problem discretely and efficiently without any relaxation. To validate the effectiveness of the proposed approach, extensive experiments on three widely used datasets are conducted and encouraging results demonstrate the superiority of ASCSH over other state-of-the-art CMH methods.