Measurements of the collision energy dependence of the chemi-ionization cross sections of the model systems He*(2 S-1,S-3) + R, R = Ar,Kr,Xe, are presented for the relevant energy range 0.003-6 eV. Except for He*(2 S-1) + Xe, all systems show a pronounced minimum of ionization at thermal energies. In the hyperthermal range, however, the two spin systems are in sharp contrast to each other: the triplet systems exhibit a broad saturating maximum of ionization, whereas the singlet systems reveal a shallow shaped second minimum (pronounced for Xe) which can be rationalized within the one-electron model potential calculations of Siska [J. Chem. Phys. 71, 3942 (1979)]. Using an effective single-channel Schrodinger equation, one finds, for all systems, complex potentials depending on distance only which properly model the measured features. A more profound picture appears when differential elastic cross sections from experiments of other groups are included. In the triplet case, differential elastic and chemi-ionization data can be adequately described with one complex potential, i.e., the local approximation holds. This is not found for the singlet systems. The hyperthermal minima instead give evidence that the optical singlet potentials are nonlocal. Various aspects of this striking spin dependence of the characteristics of He*(2 S-1,S-3) + R are discussed. In particular, the breakdown of the local approximation in the singlet case can be rigorously derived.