Balsa is very slippery, and provides no feedback. It's like having a stone for cheap, but you can screw up on it quickly. I would imagine compounds on a balsa used with a jig would match performance of any stone.
The rough side of leather attacks very quickly, but sloppily and can round an edge, plus it doesn't give consistent feedback, so it's not helpful for detecting burrs.
Smooth leather is the best for stropping, because it lets you feel the burrs and you can even see where bigger burrs are on the right leather, if your compound is clear, because it'll lightly scratch the surface, like your fingernails do. It is soft enough that it forgives(I usually try to err on the side of acute with leather), and provides good "draw" which aids in the speed with which the strop polishes. I've never used the high-end, hand worked, fennel and carrot-scrubbed, Kobe Wagyu calf ballskin that everyone seems to love, so all leather has been about the same for me.