TITLE: What is the meaning of $(h^{11},h^{21})\to (h^{11}-240,h^{21}+240)$ in Calabi-Yau threefolds?
QUESTION [36 upvotes]: 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?

REPLY [16 votes]: In the mathoverflow answer
Today's world record on the Betti numbers of Calabi-Yau three-folds.,
the paper http://arxiv.org/abs/arXiv:1207.4792 is cited. In this paper, the shift of Hodge numbers by (-240,240) is mentionned and the symmetry is explained, at least for the Calabi-Yau
threefolds obtained from reflexive polytopes by the Batyrev's construction. E_8 plays indeed an important role (via Calabi-Yau which are K3 fibrations).
EDIT: a video conference of P.Candelas on this paper : https://www.youtube.com/watch?v=4996MUz25vg