I don't get it. What does momentum have to do with the force associated with...

Force is "rate of change of momentum".

What graham said is right. Think of two charged particles, and imagine they're motionless and separated by distance x. Now just imagine that one of the particles is fixed. It's going to push the other one away with some force, and that will change its momentum. If you then repeat this scenario when the other particle has some initial momentum, it won't make any difference. At distance x, the rate of change of momentum depends upon mass, charge, and separation. It doesn't matter whether the other particle has that initial momentum, or on its direction.

Another way of looking at it is to think of a mass bouncing back and forth on the end of a spring. If that symmetry wasn't there, it would go faster one way than the other, and instead of keeping on bouncing, it would go runaway.