  We classify contravariant pairings between standard Whittaker modules and
Verma modules over a complex semisimple Lie algebra. These contravariant
pairings are useful in extending several classical techniques for category
$\mathcal{O}$ to the Mili\v{c}i\'{c}--Soergel category $\mathcal{N}$. We
introduce a class of costandard modules which generalize dual Verma modules,
and describe canonical maps from standard to costandard modules in terms of
contravariant pairings. We show that costandard modules have unique irreducible
submodules and share the same composition factors as the corresponding standard
Whittaker modules. We show that costandard modules give an algebraic
characterization of the global sections of costandard twisted Harish-Chandra
sheaves on the associated flag variety, which are defined using holonomic
duality of $\mathcal{D}$-modules. We prove that with these costandard modules,
blocks of category $\mathcal{N}$ have the structure of highest weight
categories and we establish a BGG reciprocity theorem for $\mathcal{N}$.
