A symmetrical core has rotational symmetry about the pin, not necessarily in all planes. For an object to have complete symmetry about the x, y and z axes I'm pretty sure it'd be a perfect sphere, which very few cores are.
Asymmetrical cores are note rotationally symmetrical about the core and have an imbalance in mass, hence the marked Mass Bias position where the effect of this imbalance is greatest.
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Reporting from England
