Autonomous navigation in unstructured outdoor environments is inherently challenging due to the presence of asymmetric traversal costs, such as varying energy expenditures for uphill versus downhill movement. Traditional reinforcement learning methods often assume symmetric costs, which can lead to suboptimal navigation paths and increased safety risks in real-world scenarios. In this paper, we introduce QuasiNav, a novel reinforcement learning framework that integrates quasimetric embeddings to explicitly model asymmetric costs and guide efficient, safe navigation. QuasiNav formulates the navigation problem as a constrained Markov decision process (CMDP) and employs quasimetric embeddings to capture directionally dependent costs, allowing for a more accurate representation of the terrain. This approach is combined with adaptive constraint tightening within a constrained policy optimization framework to dynamically enforce safety constraints during learning. We validate QuasiNav across three challenging navigation scenarios-undulating terrains, asymmetric hill traversal, and directionally dependent terrain traversal-demonstrating its effectiveness in both simulated and real-world environments. Experimental results show that QuasiNav significantly outperforms conventional methods, achieving higher success rates, improved energy efficiency, and better adherence to safety constraints.