In a previous paper the authors introduced the inverse measure µ^â   of a probability measure µ on [0,1]. It was argued that the respective multifractal spectra are linked by the 'inversion formula' f^â  (a) = af(1/a). Here, the statements of Part I are put in more mathematical terms and proofs are given for the inversion formula in the case of continuous measures. Thereby, f may stand for the Hausdorff spectrum, the pacing spectrum, or the coarse grained spectrum. With a closer look at the special case of self-similar measures we offer a motivation of the inversion formula as well as a discussion of possible generalizations. Doing so we find a natural extension of the scope of the notion 'self-similar' and a failure of the usual multifractal formalism.