We present a unified group-theoretical framework for superparticle theories. This explains the origin of the “twistor-like” variables that have been used in trading the superparticle's κ-symmetry for world-line supersymmetry. We show that these twistor-like variables naturally parametrise the coset space , where is the Lorentz group SO↑ (1, d−1) and is its maximal subgroup. This space is a compacy manifold, the sphere Sd−2. Our group-theoretical construction gives the proper covariantisation of a fixed light-cone frame and clarifies the relation between target-space and world-line supersymmetries.