  We study the periodic solutions of the scalar delay equation \[
\dot{x}(t)=f(x(t),x(t-1)), \] where $f$, regular enough, satisfies even-odd
symmetries and is strictly monotone in the delayed component. Our results fully
characterize the set of periodic solutions and their stability in terms of a
time map defined for a planar ordinary differential equation. The presentation
is the uniqueness counterpart of existence results developed by Kaplan and
Yorke in 1974.
