The Physics of Preforming the Eskimo Yo-Yo

The real physics behind doing this is that we are dealing with the concept of non-uniform circular motion with two masses traveling at same speeds in opposite directions.
Non-uniform circular motion by definition is circular motion with changing speeds.    (4)

In the image below, there are two masses ( pink (bottom left) and blue (bottom right) ) and one handle ( black (center) ).

     (5)                       (6)

As you can see, the lengths of the strings are different. This is mandatory because if you have two objects ("in the x-y plane") traveling towards each other, you would end up with an inelastic collision which is undesirable for this demonstration.

Once you get the system in uniform circular rotation, you have successfully preformed how to use the Eskimo Yo-Yo.

One very important key concept to understand about this is that the initial acceleration of the positioned mass must be thrown at the same velocity and must have opposite direction otherwise youll be swinging two object in the same direction at the same speed and would not be able to generate maximum power needed to keep the tension in the strings balanced.