Antimatter does have reverse charge, but sadly, not reverse gravity. Why? We should follow the standard presentation to address this question. There were prior reasons to suspect that negative energy solutions are important, but it wasn’t until Paul Dirac came along that they became indisputably necessary. What people are familiar with is Einstein’s mass-energy relation, the famous [latex]E=mc^2[/latex]. Sadly, this equation is only true for particles at rest, but there are easy alterations to make it always true in the special theory of relativity. The easiest is to just multiply the right hand side by gamma, γ, to clean up the effect of motion.
But, the most useful version of this equation for particle physics is actually [latex]E^2-p^2=m^2[/latex] where I have set [latex]c=1[/latex], the speed of light, or rather the spacetime conversion factor. In this equation, [latex]E[/latex] is the energy, [latex]p=gamma mc[/latex] is the linear momentum we can measure, and [latex]m[/latex] stands for the rest mass. In that equation, we can see that [latex]E = pm sqrt (m^2 + p^2)[/latex], and there are these annoying negative energy solutions that we had been throwing away until Dirac.
In thermodynamics, we realised that every system wants to settle to a lower state of energy. Negative energy is then a gigantic problem! We are saved by the wave-particle duality of quantum mechanics. We express a wave by [latex]exp ( i k x – w t )[/latex], which in nice units is also [latex]exp( i p x – E t )[/latex]. In a wave, the negative sign between [latex]px[/latex] and [latex]Et[/latex] represents a rightward moving wave. When the energy goes negative, then we have [latex]px + |E|t[/latex], which is a leftward moving wave.
Changing the time from [latex]t -> -t[/latex], we obtain the picture that these solutions are merely normal particles going in the opposite direction in space, and going back in time, with positive energy. Along with it, the charge on the particle is also reversed. After the discovery of antimatter, the scheme was improved until we realised that quantum field theory, most importantly quantum electrodynamics, will only work if we include these ideas. So, to summarise: We first found out negative energy solutions, and then realised that they are positive energy, but backward in time solutions.
Now, we can talk about gravity. In Einstein’s general theory of relativity, gravity—or the curvature of spacetime—is actually caused by energy and momentum. If we had negative energy, then we would have antigravity. But positive energy going backwards in time is still positive energy, and that is normal gravity. To be fair, a real test would be to gather a gigantic amount of antimatter, but the reader can see how difficult that would be. Until then, we can just continue with the successes of particle physics and accept that the standard interpretation of physics right now simply disallows antigravity.