Asymmetrical cores are cores which are NOT radially symmetric down the long axis (Low RG), which is the axis with the Pin as its end point. If we were to take cross sections down the long axis, the cross sections would NOT be symmetric.
If there are axes of symmetry in the cross section, each one would give a different shape. For instance, an equilateral triangle would be considered symmetric, because it has three identical axes of symmetry. A perfect square would also be considered symmetric, with four axes of symmetry, with two identical pairs. However, an ellipse would be asymmetric, since it has two axes of symmetry, which both look different. A regular, non-square, rectangle would also be considered asymmetric, since it also has two, non-identical axes of symmetry.
The cores in the Fury and Twisted Fury (both manufactured by Brunswick) are a good example of real world cores showing the subtle difference between symmetric and asymmetric cores.