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We consider the behaviour of odd-parity perturbations of those self-similar Lemaître-Tolman-Bondi spacetimes which admit a naked singularity. We find that a perturbation which evolves from initially regular data remains finite on the Cauchy horizon. Finiteness is demonstrated by considering the behaviour of suitable energy norms of the perturbation (and pointwise values of these quantities) on natural spacelike hypersurfaces. This result holds for a general choice of initial data and initialdoi:10.1088/0264-9381/28/10/105020 fatcat:euhdbnephzgbrc5ofz2giq5ffa