There is a good answer already. But I have some deeper thoughts:
Snakes are possibilities. They are diversified, and classified over a huge range.
Just as per their physical parameters, there are other aspects to differentiate.. like habits, habitat, and another aspect that catches lesser thoughts is Locomotion.
We all know some snakes move fast (although a very relative term with no reference), some don't.
If I ask a person how did it go? He/She would most likely reply bluntly that "It glided!"
Barring the locomotion of aquatic snakes (Hydrophiinae) and sticking strictly to the terrestrial locomotion, we can say that they have different locomotion types. Usually the type of locomotion depends on the species and what is it doing.
Very common. Almost every other snake we snake, does a Side-winding locomotion. Do not compare extremities by getting confused with the term Side-Winder. Have you seen a snake moving in mud or sand which disables a snake push against an irregular surface, like stones, sticks, etc?
In side-winding, head and tail are the supports for the snakes movement to push against. Its body is lifted off the ground to move sideways. It keeps on head and tail into the same cross direction of its body.
Observing carefully, you'll see that parts of its body positioned in one direction are never lifted off the ground, while the other segments are lifted up, resulting in a equidistant pattern drawn on the sand/mud. So, its almost half the friction of its body against the surface, lesser work.
These are also plenty around. Have you seen a snake climbing a tree or moving on a slippery surface? In this method the snake winds up itself and then it uses the energy to thrust forward. Then the front part of the body stays on the surface and the back part of the body is pulled up to be coiled again.
To notice, this method is a lot of friction with the surface, using a lot of its energy. So, do these snake move fast? No, but can they strike fast, Yes.
Have you seen a Rat snake move, or a Cobra? In this method there is movement that is from side to side. Head is lifted up, move its upper body to 10 o'clock, pull the body, oh, gotta go 2 o'clock, move its upper body that way, and repeat, the poor tail is just dragged on straight.
Have you seen a large viper move? Or a Boa, or a Python. Thats rectilinear movement. Just straight drag of the whole body. Its usually in large snakes. They just move straight.
To answer the real question: Is the skin on a snake's underside tougher than on its topside?
I think the whole answer above makes it pretty clear its less about the scales and its toughness, but more about how the snake moves.
Yet to literally answer the questions, I am really bumped up, can't really tell if there is one theory that can be applied to all the snakes. The underside scales (Subcaudal, Ventral and less significantly the paraventral scales) are more flexible, agile than the upperside scales, thats for sure.