There is a 222Rimmed now, a later Australian cartridge used mostly in re-barreled Martini Cadets. Your rifle will likely work with rimless cases. The 5.6x50R was developed by DWM, at the request of Heym because they had trouble with their Mod 22S extracting 222 Remington Magnum cases. The "Magnum" was very popular, at the time for Roe hunting, and they wanted a cartridge that was rimmed and ballistically equal. The first version was a 5.6x47R but they decided on a 50 mm case for advertising reasons. This is why the 5.6x50R was loaded with a 50-gr bullet, like the Magnum, instead of the 55-gr bullet of the later and also popular .223 Rem. Walter Grass was the Meister in "Hunting Division" of Heym at the time this was going on, but I don't remember him saying they had trouble with the Mod 22S in 222 like they did with the Magnum. The rimmed designation would be assumed.
Mike