  We examine two setups that reveal different operational implications of
path-phase complementarity for single photons in a Mach-Zehnder interferometer
(MZI). In both setups, the which-way (WW) information is recorded in the
polarization state of the photon serving as a "flying which-way detector". In
the "predictive" variant, using a {\em fixed} initial state, one obtains
duality relation between the probability to correctly predict the outcome of
either a which-way (WW) or which-phase (WP) measurement (equivalent to the
conventional path-distinguishibility-visibility). In this setup, only one or
the other (WW or WP) prediction has operational meaning in a single experiment.
In the second, "retrodictive" protocol, the initial state is secretly selected
{\em for each photon} by one party, Alice, among a set of initial states which
may differ in the amplitudes and phases of the photon in each arm of the MZI.
The goal of the other party, Bob, is to retrodict the initial state by
measurements on the photon. Here, a similar duality relation between WP and WW
probabilities, governs their {\em simultaneous} guesses in {\em each
experimental run}.
