The spin-orbit interaction for the P-state continuum of heavy alkali metals was investigated in a photoionization experiment using spin-polarized alkali atoms and circularly polarized light. From the asymmetry in ion-counting rates corresponding to the two photon helicities, Fano's spin-orbit perturbation parameter x was determined over a range of several hundred angstroms for K, Rb, and Cs. The spin-orbit perturbation was found to increase from K to Rb to Cs as expected, and the nonlinear behavior of x as a function of the photon energy E was demonstrated for K. Knowledge of x(E) was used to establish accurate values for the position of the Cooper minimum and to estimate the magnitude of the cross section at the minimum. In addition, the x(E) data for Cs were used to gain information about the spin polarization of photoelectrons in a Fano-type polarized electron source. Finally, extrapolation of x(E) for cesium into the discrete spectrum indicated the existence of a pole in the function Rho (E) which corresponds to the doublet line-strength ratio Rho (EnP)=S(nP3/2)/S(nP1/2) at the discrete energies EnP. According to our extrapolation, the pole lies in the region of n=10 to 15, in agreement with the early spectroscopic work of Sambursky (1928) and Beutell (1939), whose measurements were discounted by later investigators.