I've moved these posts about the Cavalry saddle & rifle because I'm finding it fascinating, but a bot off the original topic.
I found some more modern pictures of re-creationists in US Cavalry outfits, with their sabre tucked under the left leg in much the same place as the rifle in Richard's third picture.
Re-creationists (is this the right term??) tend to be accurate because they's that they do. These guys didn't have rifles or carbines at all, as far as I could see.
Of course it's entirely possible that different traditions grew up in different regiments - and as Richard highlights, not all mounted troops are the same anyway.
In the UK, Dragoons were the troops that rode to battle on a horse, but got off to actually fight. A Dragoon's horse would be a much cheaper and less battle-trained horse than a full cavalry mount.
The full cavalry would usually be reserved for breaking an enemy line with a mounted charge, and would charge with a sabre drawn, and possibly a pistol handy.
I really need to read up more about this. As with the Romans, Native Americans, Ships and the Knights, the Playmobil theme of US Civil War and the US Cavalry is moving me to do more research (if only a little light reading with lots of colour pictures!).