A local magnetic equilibrium solution is sought around the magnetic axis in order to identify the key parameters defining the magnetic-surface's up-down asymmetry in the core of tokamak plasmas. The asymmetry is found to be determined essentially by the ratio of the toroidal current density flowing on axis to the fraction of the external field's odd perturbation that manages to propagate from the plasma boundary into the core. The predictions are tested and illustrated with experimentally relevant numerical equilibria. Hollow current-density distributions, and hence reverse magnetic shear, are seen to be crucial to bring into the core asymmetry values that are usually found only near the plasma edge.