  We introduce the notion of a \emph{Whitney dual} of a graded poset. Two
posets are Whitney duals to each other if (the absolute value of) their Whitney
numbers of the first and second kind are interchanged between the two posets.
We define new types of edge and chain-edge labelings which we call
\emph{Whitney labelings}. We prove that every graded poset with a Whitney
labeling has a Whitney dual. Moreover, we show how to explicitly construct a
Whitney dual using a technique involving quotient posets.
  As applications of our main theorem, we show that geometric lattices, the
lattice of noncrossing partitions, the poset of weighted partitions studied by
Gonz\'alez D'Le\'on-Wachs, and most of the R$^*$S-labelable posets studied by
Simion-Stanley all have Whitney duals. Our technique gives a combinatorial
description of a Whitney dual of the noncrossing partition lattice in terms of
a family of noncrossing Dyck paths. Our method also provides an explanation of
the Whitney duality between the poset of weighted partitions and a poset of
rooted forests studied by Reiner and Sagan. An integral part of this
explanation is a new chain-edge labeling for the poset of weighted partitions
which we show is a Whitney labeling.
  Finally, we show that a graded poset with a Whitney labeling admits a local
action of the $0$-Hecke algebra of type $A$ on its set of maximal chains. The
characteristic of the associated representation is Ehrenborg's flag
quasisymmetric function. The existence of this action implies, using a result
of McNamara, that when the maximal intervals of the constructed Whitney duals
are bowtie-free, they are also snellable. In the case where these maximal
intervals are lattices, they are supersolvable.
