  We study multi-photon resonances in a strongly-driven three-level quantum
system, where one level is periodically swept through a pair of levels with
constant energy separation $E$. Near the multi-photon resonance condition
$n\hbar\omega = E$, where $n$ is an integer, we find qualitatively different
behavior for $n$ even or odd. We explain this phenomenon in terms of families
of interfering trajectories of the multi-level system. Remarkably, the behavior
is insensitive to fluctuations of the energy of the driven level, and survives
deep into the strong dephasing regime. The setup can be relevant for a variety
of solid state and atomic or molecular systems. In particular, it provides a
clear mechanism to explain recent puzzling experimental observations in
strongly-driven double quantum dots.
