I've trained all three of my cats. They can and do respond very well to positive feedback (usually a small treat or being petted). As opposed to a dog which you can "handle", i.e., by pushing down on their hindquarters, lifting a paw, etc., with a cat you have to wait for them to perform some part of the trick you want. Then you reinforce that behavior. After they learn the "partial trick" you keep adding on until they have the entire thing down. It takes a lot more patience and a lot longer. Also, cats will "revert" and decide to do some simpler thing, which forces you to start all over.
I'd have to say, that while there are some smart cats, dogs are definitely better learners and much more engaging. In my experience, dogs pick up & retain new behaviors a lot better than cats (given that I've only had three dogs & three cats, take this as the anecdotal evidence it is). I don't really know what you mean by "communicate with them on a fairly deep level" ... sounds a little ... mystical to me. My cats love attention, love food, and love to be "trained" (which is a combination of the previous two).