It was entirely horse cavalry, but not really intended to fight as such, with lances and sabres and whatnot. Instead, the proper 18th or 19th Century term would have been “Dragoons.” They were armed with rifles, usually light carbines, and in the event of a firefight, intended to dismount and fight on foot as light infantry. By 1939, the whole horse thing was intended for mobility to the battlefield, not on it.
Sounds like mounted infantry, They were doing this in the US Civil War, not a good idea against tank cavalry.