US cavalry at the time (1940) was still partially mounted. Cavalry reconnissance squadrons were semi-mechanized. Part rode on M3 scout cars the other part used horses. More weird still, the horsed portion had trucks and trailers provided to move their horses in non-tactical moves as they could not otherwise keep up with the mechanized portion of the unit.
In addition, there was still a cavalry division in existance and a second forming. Both were fully mounted. Neither ended up seeing combat mounted. The First went to the Pacific as dismounted infantry and the Second was broken up for replacements in North Africa.