How is the HR process not symmetric? Whatever would cause the dissolution of the black hole--how would the same process happening in reverse (matter falling into the black hole and anti-matter escaping) not cause equilibrium to be maintained?
Matter and anti-matter annihilating each other produces energy. That energy remains trapped in the hole, so it's mass is unchanged (energy = mass * lightspeed squared). So whether matter or anti-matter fall into the hole, its mass will grow.
What's happening with Hawking radiation is (AFAIK) that particles are actually ripples in a field - for example, a photon is a ripple in the electromagnetic field, and an electron is a ripple in the "electron field". If these ripples are well-formed waves they're said to be "real" and have measurable quantities, such as energy. Otherwise they're "virtual" particles. That's what's meant when they say electromagnetism works by exchanging virtual photons: charged particles cause disturbances in the electromagnetic field, which affect other charged particles.
Now, quantum mehcanics has a number of paired properties, where both members of the pair can't have an exact value at the same time. Position and momentum are perhaps the most famous such pair, but another is the value and rate of change of a field. That means that no field's value can be permanently zero, because then we'd know both the exact value (0) and exact rate of change (also 0). And because they can't be permanently 0 despite having to average to it, they must ripple randomly, a state of affairs often referred to as "quantum foam". Since particles are ripples in these fields, this process is usually depicted as a pair of virtual particles coming from nowhere, living for a while, and then meeting again and canceling each other out.
An interesting property of these virtual particle pairs is that if they receive at least as much energy as corresponds to the pair's rest mass, for example by being accelerated in a black hole's gravity field, they become "real". That is, random ripple turns into a well-defined wave that's not going to disappear, but is going to continue its existence as a real particle. According to current cosmological theories, this is how all matter originated, with the Big Bang acting as a source of energy. This is also how particle accelerators work: smash a stream of particles together hard, so the energy of the collision makes virtual particles materialize.
So what happens is that if a pair of virtual particles is brought to existence near the even horizon of a hole, one might fall in and the other fly away. The one flying outwards loses energy - it's rising against gravity, after all - but the one falling in gains more than enough to offset that, since it gets pulled harder and harder the farther it goes. So, the pair of particles as a whole gain energy, and if that energy exceeds the pair's rest mass the pair turns into real particles. Since energy must be conserved, it must come from somewhere, and the only available source is the black hole's mass, which is diminished as a result. The falling particle is basically carrying a bill with it, which negates some of the black hole's energy. And since it's the direction of gravity that determines which particle gets the bill, it's always the one falling into the hole, making the process asymmetric.
But then again, the actual math is beyond me, so I could be completely wrong here.