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This paper investigates tail-biting trellis realizations for linear block codes. Intrinsic trellis properties are used to characterize irreducibility on given intervals of the time axis. It proves beneficial to always consider the trellis and its dual simultaneously. A major role is played by trellis properties that amount to observability and controllability of trellis fragments of various lengths. For fragments of length less than the minimum span length of the code it is shown that fragment observability and fragment controllability are equivalent to irreducibility. For reducible trellises, a constructive reduction procedure is presented. The considerations also lead to a characterization for when the dual of a trellis allows a product factorization into elementary (“atomic”) trellises.