We explore a continuum of observably and usefully inequivalent, finite-dimensional off-shell representations of worldline N=4-extended supersymmetry, differing from one another only in the value of a tuning parameter. Their dynamics turns out to be n
ontrivial already when restricting to just bilinear Lagrangians. In particular, we find a 34-parameter family of bilinear Lagrangians that couple two differently tuned of these supermultiplets to each other and to external magnetic fields, where the explicit dependence on the tuning parameters cannot be removed by any field redefinition and so is observable.
