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Many graphics and computer-aided design applications require that the polygonal meshes used in geometric computing have the properties of not only 2-manifold but also are orientable. In this paper, by collecting previous work scattered in the topology and geometry literature, we rigorously present a theoretical basis for orientable polygonal surface representation from a modern point of view. Based on the presented basis, we propose a new combinatorial data structure that can guarantee the property of orientable 2-manifolds and is primal/dual efficient. Comparisons with other widely used data structures are also presented in terms of time and space efficiency.