  Motivated by the roll-switching behavior observed in rotating
Rayleigh-B\'enard convection, we define a K\"uppers-Lortz (K-L) state as a
volume-preserving flow with periodic roll switching. For an individual roll
state, the Lagrangian particle trajectories are periodic. In a system with
roll-switching, the particles can exhibit three-dimensional, chaotic motion. We
study a simple phenomenological map that models the Lagrangian dynamics in a
K-L state. When the roll axes differ by $120^{\circ}$ in the plane of rotation,
we show that the phase space is dominated by invariant tori if the ratio of
switching time to roll turnover time is small. When this parameter approaches
zero these tori limit onto the classical hexagonal convection patterns, and, as
it gets large, the dynamics becomes fully chaotic and well-mixed. For
intermediate values, there are interlinked toroidal and poloidal structures
separated by chaotic regions. We also compute the exit time distributions and
show that the unbounded chaotic orbits are normally diffusive. Although the map
presumes instantaneous switching between roll states, we show that the
qualitative features of the flow persist when the model has smooth, overlapping
time-dependence for the roll amplitudes (the Busse-Heikes model).
