Brazil has a wide variety of riding styles, and the saddles and tack vary quite a bit regionally. Here is a quick look at two kinds of saddles used in southern Brazil. The basto sela is a particularly interesting one - the two sides of the saddle are held together by stitching, making it a very flexible saddle. The "four headed" kind mentioned in the video is split all the way through the pommel and cantle, making it even more flexible. There is an image of one on this site, and you can see how the pommel and cantle each consist of two unconnected sections (thus making the four "heads" in the name).
Here is an interior photo from a tack shop in the gaucho region of the south, that shows a variety of saddles, including a fancy red sidesaddle on the upper left, and a dressy white one in the center. It is typical to store saddles by hanging them on pegs or hooks, like they are here.
Finally, here is a short video showing a saddle be adjusted on a horse's back. You can see the way the tack is layered on - first the thick wool pads, then the leather saddle (notice that the stirrups are attached externally), then a girth to hold the saddle (the girth is not sewn onto the saddle in any way), then the sheepskin, then a second girth to hold the sheepskin in place.
Thanks, Paulo, for the explanations and demonstrations!