For the gauge massless higher spin 4D, Script N = 1 off-shell supermultiplets previously developed, we provide evidence of a twistor-like oscillator realization that is intrinsically related to the superfield structure of the dynamical variables and gauge transformations. Gauge invariant field strengths and linearized Bianchi identities for these multiplets are worked out. It is further argued, inspired by earlier non-supersymmetric constructions due to Klishevich and Zinoviev, that a massive superspin-s multiplet can be described as a gauge-invariant dynamical system involving massless multiplets of superspins s,s−1/2,...,0. A gauge-invariant formulation for the massive gravitino multiplet is discussed in some detail.