Comment Origin of that statement (Score 1) 122
Symmetry in this case refers to this: If you take a particle or a diagram describing a particle interaction and "flip something", you get something new that is still valid. Take a proton and flip the charge and you get an anti-proton for instance. Because of this symmetry, matter and anti-matter behave in exactly the same way, or so we belive. your particle and your the flipped version decay in the same way for instance.
What physicicst discovered over the last century was that it's not enough to flip the charge to make this valid in all cases. You have to flip charge, direction of time and flip something called "parity". Flipping the direction of time means simply to draw a feynman diagram of a particle collision or interaction on an overhead foil so that time runs from left to right and then turn it backwards. That's the mirror-image of the interaction.
But as I said you have to flip both time, parity and charge to get a valid diagram, and that's the origin of the term "an antiparticle is a particle travelling backwards in time"