Cross-modal hashing retrieval encodes heterogeneous data into compact binary codes for efficient Hamming-space search. Existing methods usually learn cross-modal semantics in continuous feature spaces and generate binary codes through a final sign operation, which weakly couples training optimization with discrete hash retrieval. We propose SpikeHash, a unified spiking framework that formulates cross-modal hashing as spike-state evolution, directional spike interaction, and competitive spike readout. Specifically, SpikeHash converts image and text features into multi-timestep spike sequences. In a shared Hamming space, the two spike sequences jointly drive the temporal evolution of a shared hash state. Cross-modal interaction is further performed through directional spike modulation, enabling each modality to influence the firing dynamics of the other. Crucially, SpikeHash replaces the conventional continuous hash head with a positive-negative spiking hash readout, where each hash bit is produced by temporal competition between paired spike channels. Experimental results show that SpikeHash achieves competitive retrieval accuracy on three benchmark datasets while reducing the parameter size, operation count, and estimated energy of the hash learning stage, suggesting a compact spiking alternative to conventional continuous hash mapping. The project page is available at https://shuqiao-111.github.io/.