We study analytically the behaviour of odd-parity perturbations of those self-similar Lemaitre-Tolman-Bondi spacetimes which admit a naked singularity. We find that a perturbation which evolves from initially regular data remains finite on the Cauchy hori- zon. This holds for gauge invariant matter and metric perturbations, where 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. Copyright © 2012 by World Scientific Publishing Co. Pte. Ltd.