Super-gauge transformations on physical space-time are defined on the projective six-dimensional hypercone. The basic observation which motivates the present approach is that the product of any two super-gauges is a general conformal (+ γ₅) transformation and indeed these latter transformations are simply realized on the six-dimensional hypercone. It turns out that also super-gauge transformations have a natural action on this projective space where the parameters of such transformations are (totally anticommuting) spinors which transform according to the fundamental (eight-dimensional) representation of the spinor group SU (2,2), isomorphic to the conformal group. The enlarged algebra of super-gauges, conformal and chiral transformations and its most relevant representations are discussed in the present formalism.