By browsing through the Hodge data of known Calabi-Yau threefolds, I stumbled upon an observation that frequently enough a pair of Hodge numbers $(h^{11},h^{21})$ comes together with the pair $ (h^{11}-240,h^{21}+240)$. This shift, together with mirror symmetry accounts for a certain symmetry in the top portion of the Kreuzer-Skarke plot.

Has anyone observed this shift before?

Is there an empirical explanation for this shift? Is there any relation to the roots of $E_8$?

Are there other similar but smaller shifts?

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