  It was shown by Kuznetsov that complete intersections of $n$ generic quadrics
in ${\mathbb P}^{2n-1}$ are related by Homological Projective Duality to
certain non-commutative (Clifford) varieties which are in some sense birational
to double covers of ${\mathbb P}^{n-1}$ ramified over symmetric determinantal
hypersurfaces. Mirror symmetry predicts that the Hodge numbers of the complete
intersections of quadrics must coincide with the appropriately defined Hodge
numbers of these double covers. We observe that these numbers must be different
from the well-known Batyrev's stringy Hodge numbers, else the equality fails
already at the level of Euler characteristics. We define a natural modification
of stringy Hodge numbers for the particular class of Clifford varieties, and
prove the corresponding equality of Euler characteristics in arbitrary
dimension.
