The excitation function and angular distributions of elastic alpha-particle scattering on Si-28 have been measured in the laboratory energy range 6-28 MeV using a backscattering technique on a thick target, yielding a continuous energy distribution. More than 200 narrow states are observed, with widths in the range similar to 30-100 keV at excitation energies E* = 13-32 MeV. Angular distributions at backward angles were measured, and angular momentum values of more than 83 states have been deduced. The analysis gives spin-parities J(pi), alpha-partial widths Gamma(alpha) and reduced widths of the narrow high-lying resonant states in S-32. The experimentally observed states display both the negative- and the positive-parity rotational-like sequences with seemingly no parity splitting, a finding which is at variance with most potential-model predictions. The deduced effective moment of inertia indicates a more extended structure than the ground-state configuration. The observed strength of each l-value is analyzed in terms of an underlying split doorway state of Lorentzian form, which yields an interpretation as fragmented rotational alpha + Si-28 states.