An analysis is made of the Stokes flow between parallel planes due to a three-dimensional rotlet whose axis is parallel to the boundary planes. The separation in the plane of symmetry of this flow is compared with that in its two-dimensional analogue, the Stokes flow between parallel planes due to a two-dimensional rotlet. It is found that when the rotlets are midway between the planar walls, both flows exhibit an infinite set of Moffatt eddies. However, when the rotlets are not midway between the walls, the two-dimensional flow has an infinite set of Moffatt eddies, while the three-dimensional flow has at most a finite number of eddies and behaves, far from the rotlet, like the flow due to a two-dimensional source–sink doublet in each of the planes parallel to the boundary planes. An eigenfunction expansion describing a class of asymmetric Stokes flows between parallel planes is also derived and used to show that the far-field behaviour of flows in this class generally resembles the aforementioned flow due to a two-dimensional source–sink doublet in the planes parallel to the walls.