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We study systems with parity invariant contact interactions in one dimension.The model analyzed is the simplest nontrivial one --- a quantum wire with apoint defect --- and yet is shown to exhibit exotic phenomena, such as strongvs weak coupling duality and spiral anholonomy in the spectral flow. Thestructure underlying these phenomena is SU(2), which arises as accidentalsymmetry for a particular class of interactions.Comment: 4 pages ReVTeX with 4 epsf figures. KEK preprint 2000-3. Correction  in Eq.(14

Duality and Anholonomy in Quantum Mechanics of 1D Contact Interactions