<p>Bipartite networks appear in many real-world contexts, linking entities across two distinct sets. One-mode projections, though widely used, can introduce artificial correlations and inflated clustering, obscuring the true underlying structure. In this work, we propose a geometric model for bipartite networks that leverages the high levels of bipartite four-cycles as a measure of clustering and embeds both node types into a shared similarity space, where the link probability decreases with distance. Additionally, we introduce B-Mercator, an algorithm that infers node positions from the bipartite structure. We evaluate its performance on diverse datasets, illustrating how the resulting embeddings improve downstream tasks such as node classification and distance-based link prediction. These hyperbolic embeddings enable realistic synthetic network generation with node features mirroring real data. By preserving the bipartite structure, our approach avoids projection biases, offering more accurate structural descriptions and providing a robust framework for uncovering hidden geometry in bipartite networks.</p>