We have studied the three-component liquid crystal mixture reported by Inui et al. to exhibit `thresholdless antiferroelectricity', as well as some other materials with the same V-shaped (linear) electro-optic switching behavior. We find that thresholdless behavior is obtained in the absence of antiferroelectricity. The characteristic response of these materials is due to the field-induced switching of a twisted smectic C* structure stabilized by strong polar surface interactions. The twist is confined to thin surface regions by polarization charge effects leaving the bulk of the cell uniform, which gives almost complete extinction at zero field. Model calculations of the electro-optic response are in good agreement with the experimental data. In sufficiently thin cells, such thresholdless switching can in fact be followed down to much lower temperatures, where the bulk would be antiferroelectric but the cell is maintained in the ferroelectric state by hysteresis.