We consider here passive mechanical wrists, capable of imparting a desired damping matrix to a grasped workpiece. Previous work [12, 13] has shown how to select a damping matrix such that an assembly operation can be made force-guided. The passive mechanical wrist is to be programmable - it can adopt a wide range of damping matrices - by virtue of a number of tunable dampers which interconnect the joints. We have been studying the range of damping matrices that such a wrist can adopt, purely by tuning its dampers. We find that a redundant wrist has a broader range of realizable damping matrices than a non-redundant wrist. A kinematic Jacobian relates the task-space damping matrix to a similar matrix in the hydraulic space of the tunable dampers (joint-space). For redundant wrists the transformation of damping matrices between task-space and joint-space is not straightforward. In this paper we identify the casual directions along which the transformations are linear. We show that the joint-space matrices which are obtained as linear transformations of desired task-space matrices are all singular. Many realizable joint-space matrices (corresponding to a desired task-space damping matrix) are shown to exist which are not discovered by linear transformations.