  A necessary and sufficient condition is obtained for the existence of strong
stationary times for ergodic one-dimensional diffusions, whatever the initial
distribution. The strong stationary times are constructed through intertwinings
with dual processes, in the Diaconis-Fill sense, taking values in the set of
segments of the extended line $\mathbb{R}\sqcup\{-\infty,+\infty\}$. They can
be seen as natural $h$-transforms of the extensions to the diffusion framework
of the evolving sets of Morris-Peres. Starting from a singleton set, the dual
process begins by evolving into true segments in the same way a Bessel process
of dimension 3 escapes from 0. The strong stationary time corresponds to the
first time the full segment $[-\infty,+\infty]$ is reached. The benchmark
Ornstein-Uhlenbeck process cannot be treated in this way, it will nevertheless
be seen how to use other strong times to recover its optimal exponential rate
of convergence in the total variation sense.
