  We show that there exists a natural non-degenerate pairing of the
homomorphism space between two neighbor standard modules over a
quasi-hereditary algebra with the first extension space between the
corresponding costandard modules and vise versa. Investigation of this
phenomenon leads to a family of pairings involving standard, costandard and
tilting modules. In the graded case, under some "Koszul-like" assumptions
(which we prove are satisfied for example for the blocks of the category
$\mathcal{O}$), we obtain a non-degenerate pairing between certain graded
homomorphism and graded extension spaces. This motivates the study of the
category of linear tilting complexes for graded quasi-hereditary algebras. We
show that, under assumptions, similar to those mentioned above, this category
realizes the module category for the Koszul dual of the Ringel dual of the
original algebra. As a corollary we obtain that under these assumptions the
Ringel and Koszul dualities commute.
