Cavorite, as described in HG Wells' First Men In The Moon, was an alloy that shielded materials from gravity. It was thus a metamaterial that acted as a gravity cloak, just as we are currently working on invisibility cloaks to hide objects or events, and magnetic cloaks to create an antimagnet.
Upsidaisium, on the other hand, was a gravity repellent. It consisted of a material (a transmaterial? a material that changes both itself and the surrounding environment?) that was a gravity repulsor, or rather, was a repulsor to normal matter. It fell up.
Is there such a thing? Possibly. Scientists at CERN and other places are trying to determine if antimatter has positive or negative mass. We've never produced enough, or kept enough of it around long enough, to know. But that has now changed. Scientists have been able to keep the stuff in magnetic bottles for up to fifteen minutes. Considering the storage time used to be only a few thousandths of a seconds as recently as last year, that's a considerable accomplishment. And now, an experiment at CERN called AEGIS may soon give us an answer as to whether antihydrogen falls up or down - possibly by October of 2011.
The Dirac equation (derived by Paul Dirac) has a solution that predicts antiparticles. It actually has four solutions, and two of them are for particles with negative energy, and thus negative mass. If it turns out that antiparticles do have negative mass, and are thus repulsive to regular matter, then this could be a very big thing.
First of all, it could mean that dark matter - that matter that cannot be seen but is felt gravitationally, in the faster rotation of galaxies - may not actually be there. Dark matter may be an illusion caused by the quantum vacuum. CERN physicist Slavkov Hajdukovic, a dark matter skeptic, has proposed a theory whereby virtual particles (that can briefly pop out of the vacuum, and then merge and annihilate themselves back into nothingness) are gravitational dipoles (a normal particle of mass and an antiparticle of negative mass), then the gravitational field generated will be stronger than predicted, and thus look like a dark matter particle - not seen but felt. It would certainly simplify things. All other things being equally messy, the principle of parsimony is rather preferred.
Ah, but it goes further than that. One of the more exotic speculations in physics is the creation of wormholes, funny spacelike paths that can link different regions of the universe without crossing through. One possible way to create a stable wormhole is with the use of a strange material that has negative mass. A ring of such a material could keep a wormhole mouth from collapsing. Of course, this is all rank speculation, but it could be something to keep an eye on.
Because, if you can create a wormhole, you can violate causality. And global causality violation is the new cupcakes.
Better start saving up your antiprotons now, fellow babies.