The RAdial Velocity Experiment survey, combined with proper motions and distance estimates, can be used to study in detail stellar kinematics in the extended solar neighbourhood (solar suburb). Using 72 365 red-clump stars, we examine the mean velocity components in 3D between 6 < R < 10 kpc and −2 < Z < 2 kpc, concentrating on north-south differences. Simple parametric fits to the (R, Z) trends for V φ and the velocity dispersions are presented. We confirm the recently discovered gradient in mean Galactocentric radial velocity, V R, finding that the gradient is marked below the plane (δ〈V R〉/δR = −8 km s−1 kpc−1 for Z < 0, vanishing to zero above the plane), with a Z gradient thus also present. The vertical velocity, V Z , also shows clear, large-amplitude (|V Z | = 17 km s−1) structure, with indications of a rarefaction-compression pattern, suggestive of wave-like behaviour. We perform a rigorous error analysis, tracing sources of both systematic and random errors. We confirm the north-south differences in V R and V Z along the line of sight, with the V R estimated independent of the proper motions. The complex three-dimensional structure of velocity space presents challenges for future modelling of the Galactic disc, with the Galactic bar, spiral arms and excitation of wave-like structures all probably playing a role.