Do Electric Charges in the Van Allen Radiation belt move in Opposite Directions?
Yes and no. The question @JohnRennie links to in his comment  Analogy between magnetic bottle and Van Allen's radiation belt contains a lot of relevant information, but packaged as an answer to this question.

The question correctly concludes that, at least in the plane perpendicular to the magnetic field, particles will travel in circles, and those circles will be in opposite directions, but they do not, generally simply circle around the Earth as you suggest. There are two reasons for this: firstly the magnetic fields are too strong, or the particles too light, so the circles you get are much smaller and secondly because they are also moving in the third direction. So you should think of them travelling in fairly tight spirals around the magnetic field lines, which, roughly speaking run between the North and South magnetic poles. 

As they approach the poles, the field gets stronger (the lines bunch up) and it turns out that this repels the particles, so they end up bouncing back and forth between the two poles (all in spirals around the magnetic field lines).  The protons and electrons do spiral in opposite directions.