Turbulence is a high-dimensional dynamical system with known equations of motion. It can be numerically integrated, but the simulation results are also high-dimensional and hard to interpret. Lower-dimensional models are more useful, but they are not dynamical systems because some dynamics is discarded in the projection. They are best described by their projected probability distribution, where the dynamical system is substituted by an evolution operator that has a deterministic component describing the behaviour in the projected subspace, and an apparently stochastic one that reflects the effect of the discarded dimensions. Using as example turbulence at moderate but non-trivial Reynolds number, we show that particularly deterministic projections can be identified by either Monte-Carlo or exhaustive testing. They can be interpreted as coherent structures. We also show that, for projections in which the deterministic component is dominant, ‘physics free’ stochastic Markovian models can be constructed that mimic many of the statistics of real flows, even for fairly crude approximations to the transition operator. Purely deterministic models fail by quickly converging to a steady state. This is illustrated by constructing data-driven reduced models for a moderate-Reynolds number turbulent channel.