There are a number of ways cleavers are divided, slicing, chopping, thin, all purpose. I find the most helpful to be by edge type, thin, medium, and thick. Equivalent gyutos would be, thin or lazer, all-purpose or work horse for medium. A Western Deba would be similar to a thick edge cleaver. The most popular cleaver type, is the thin edge, slicing cleaver.
The size and weight of a cleaver are the common complaint of people who don't care for them. Its the size and weight are what makes a cleaver work. The height of a cleaver aligns the wrist and forearm, so that the larger muscles of the forearm can be used. That's why it can be less fatiguing to use a cleaver then a gyuto.
The height acts as a edge guide. As long as the edge doesn't get lifted above the first knuckle, then it is almost impossible to get cut. It is much easier to make a straight horizontal cut with a cleaver then a gyuto, because the blade doesn't get lost in the food. Board management is easy with a cleaver. Usually I can clear the board with one pass with a cleaver, it can take up to three or four passes with a gyuto.
The biggest objection to cleaver is the weight. The weight of a cleaver assists in making the cut. Placed on top of food, the lightest pressure should make the cut. Most of the energy used with a cleaver is lifting up the blade. The movements of a cleaver are generally small, so it ends up being an efficient way to cut.
Most of the problems people have with a cleaver can be traced back to the grip. The pinch grip on a cleaver usually does not work. The blade of a cleaver is held between the forefinger and thumb. While I've seen videos of people who curl up there forefinger against the blade, I get better results when I extend my thumb and forefinger down the blade. Best control comes when I extend my index finger down the blade, sort of an upside down peace sign. In the above video Andy demonstrates both grips.
The other issue with cleavers is that people don't give themselves enough time to learn how to use them. For people who have grown up with western style knifes, the cleaver is an odd knife. It takes about a month to just get comfortable with a cleaver. Even after using cleavers for five years, I don't know if they will ever feel natural, the way a gyuto does.
I do feel that learning to use a full size cleaver 220 x 110mm, in the 400-600 gram range has improved my knife skills. All gyutos feel very small in my hands. Plus the control I learned on cleavers has transferred over to gyutos.
Jay