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?



What is the meaning of $(h^{11},h^{21})\to (h^{11}-240,h^{21}+240)$ in Calabi-Yau threefolds?
In the mathoverflow answer
[Today's world record on the Betti numbers of Calabi-Yau three-folds.](https://mathoverflow.net/questions/110535/todays-world-record-on-the-betti-numbers-of-calabi-yau-three-folds/110547#110547),
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> 


