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We present an experimental investigation into the phenomenon of gestural symmetry for two-handed gestures accompanying speech. We describe an approach to compute hand motion symmetries based on the correlation computations. Local symmetries are detected using a windowing operation. We demonstrate that the selection of a smaller window size results in better sensitivity to local symmetries at the expense of noise in the form of spurious symmetries and 'symmetry dropoffs'. Our algorithm applies a 'hole filling' post process to address these detection problems. We examine the role of the detected motion symmetries of two-handed gestures in the structuring of speech. We compared discourse segments corresponding to extracted symmetries in two natural conversations against a discourse analysis by expert psycholinguistic coders. These comparisons illustrate the effectiveness of the symmetry feature for the understanding of underlying discourse structure. We believe that this basic characteristic of two-handed gestures accompanying speech must be incorporated in any multimodal interaction system involving two-handed gestures and speech.