A $1/r^2$ force occurs because the particle mediating the force is massless. The only way there could be another such particle that had not yet been detected would be if the coupling constant for this force was small or zero for ordinary forms of matter. For example, suppose that we, our world, and our lab apparatus were all made out of neutrons. We might have a very hard time learning of the existence of the electromagnetic force. (Neutrons do have a magnetic moment, though.) I could imagine, for example, that dark matter possesses some new type of charge (not electric charge) that is zero for baryonic matter.
Re duals, these are general features of all fields due to the way special relativity works. For example, there is a standard, easy argument http://www.lightandmatter.com/html_books/0sn/ch11/ch11.html#Section11.1 that if we have electrical interactions plus SR, we must have magnetic interactions. However, the detailed nature of the dual depends on the nature of the field equations, or, in QFT terms, on the spin of the force-carrying particle. For example, the graviton has spin 2, not spin 1 like the photon, so we don't have a dual structure exactly like (E,B) in E&M. There are, however, some very nice analogies between the (E,B) structure and what you see in stationary fields such as that of the rotating earth. GR does have phenomena that people refer to as "gravitomagnetism," etc. But the analogy is not perfect, because otherwise GR would be the same theory as E&M. The fundamental reason for the difference is the spin of 2 instead of 1.