Several first-order methods for rearrangement collisions are compared by applying them all to the same problem: electron-exchange scattering off the hydrogen atom. The methods considered are Born-Oppenheimer (BO); Bates-Bassel-Gerjuoy-Mittleman (BBGM); and Ochkur (O) with its modifications by Rudge and Bely. Whenever possible, they are compared with close-coupling calculations or polarized-orbital calculations. The BO approximation gives better results if the core term is retained, indicating that this inclusion helps to achieve a cancellation of errors. The BBGM first-order method is found to give reasonable integral and differential cross sections at intermediate energies but not at low energies. The methods of Ochkur and Rudge give improved integral cross sections, but at low energies the differential cross sections may be poor. The unitarization procedure of Seaton and Bely is extended to calculate exchange scattering off neutral systems but only slightly improves the Ochkur-Rudge (OR) results. None of the methods examined is satisfactory in all regards. Several other improved methods are also reviewed.