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Geometric transitions between Calabi-Yau manifolds have proven to be a powerful tool in exploring the intricate and interconnected vacuum structure of string compactifications. However, their role in N=1, 4-dimensional string compactifications remains relatively unexplored. In this work we present a novel proposal for transitioning the background geometry (including NS5-branes and holomorphic, slope-stable vector bundles) of 4-dimensional, N=1 heterotic string compactifications through a conifold transition connecting Calabi-Yau threefolds. Our proposal is geometric in nature but informed by the heterotic effective theory. Central to this study is a description of how the cotangent bundles of the deformation and resolution manifolds in the conifold can be connected by an apparent small instanton transition with a 5-brane wrapping the small resolution curves. We show that by a "pair creation" process 5-branes can be generated simultaneously in the gauge and gravitational sectors and used to describe a coupled minimal change in the manifold and gauge sector. This observation leads us to propose dualities for 5-branes and gauge bundles in heterotic conifolds which we then confirm at the level of spectrum in large classes of examples. While the 5-brane duality is novel, we observe that the bundle correspondence has appeared before in the Target Space Duality exhibited by (0,2) GLSMs. Thus our work provides a geometric explanation of (0,2) Target Space Duality.