Why do tops move opposite to each other when colliding, not tangentially? When two well-balanced tops collide, they tend to bounce directly away from each, in other words along the line connecting their centers.
Intuitively I would expect the tops to move tangentially, not orthogonally.
What is causing them to move away from each other in this way?
 A: This pattern is due to a combination of Conservation of total linear AND angular momentum before and after collision. Especially if they have the same mass, this will be more pronounced. 
As accidentalfouriertransform pointed out, the spinning nature of tops must be considered too. Since they slow down (due to contact frictions), by conservation of total angular momentum, they must separate in a direction that will more efficiently increase their separation in order to conserve the product of moment of inertia and angular velocity of the system as a whole. This will result in a final state in which two tops will separate back-to-back in arbitrary directions in the lab frame. (The direction of separation is due to conservation of linear momentum while the magnitude of separation is due to conservation of angular momentum).
A: The important laws here are momentum conservation and angular momentum conservation.
There are two possible contributions to momentum. The momentum of the center of mass of the tops, and the instanateous momentum from the contact point of the scatter.
If the angular velocity of the top is large enough so that at its radius the speed is much larger than the center of mass momenta, the interaction will be grazing and momenta transferred according to the instantaneous speed vector at the surface of the touching tops. They will either separate tangentially if the spins are in the opposite direction  reducing their angular momentum, stop, if they are exact, or move in the almost same direction if the spins are in the same direction, getting linear momentum, and fall due to friction etc. (Two spinning satellites in the latter case would balance angular momentum with the separation between each other).
If the momentum is much larger than the effective impulse when the two spins touch due to the rotation, the direction of scatter will be in the direction of conserved momentum. If of a similar order of magnitude something in between the first and the second option, depending on the spins directions. Angular momentum will be conserved by the relative angular momentum between the tops.
Hope this is not too confusing .
