  A common theme of enumerative combinatorics is formed by counting functions
that are polynomials evaluated at positive integers. In this expository paper,
we focus on four families of such counting functions connected to hyperplane
arrangements, lattice points in polyhedra, proper colorings of graphs, and
$P$-partitions. We will see that in each instance we get interesting
information out of a counting function when we evaluate it at a \emph{negative}
integer (and so, a priori the counting function does not make sense at this
number). Our goals are to convey some of the charm these "alternative"
evaluations of counting functions exhibit, and to weave a unifying thread
through various combinatorial reciprocity theorems by looking at them through
the lens of geometry, which will include some scenic detours through other
combinatorial concepts.
