FlyboyJ - thanks for the response. I agree with everything you stated. Yes the torque links caster with the wheel freely, and yes the steering is accomplished with differential braking or rudder. The dampers are passive devices that are merely loaded up with pressure; they do not steer. But look at the picture - the damper cylinders move with the castering upper torque link; they have to as they are firmly attached to the same fitting on the strut. But the piston rod parts of the dampers have their rod ends attached to lugs that hang down from an upper strut fitting that is fixed i.e. it doesn't caster. That means as the wheel casters to and fro, the pistons of the dampers must plunge in and out of the damper cylinders, providing the damping. But the damper cylinders are firmly attached to the upper torque link strut fitting and are not allowed to wiggle or rotate or whatever you want to call that secondary motion. They just move with the whole lower part of the landing gear. If the piston rods are tight fitting in the cylinders and can only slide in and out, and if the cylinders are firmly attached to the lower strut, and if the piston rod ends each have a simple single pin attachment to the fixed upper strut lug which allows only simple rotational motion about the bolt centerline, then there is a kinematic problem with the device.
The only thing I can think of is that the forward (piston head) end of the rod, which slides on the inside surface of the cylinder, and which is a larger diameter than the rod itself just like an engine piston, is shaped such that the rod can wiggle side to side as well as in and out, and there is clearance at the aft end of the cylinder for teh rod to do so. But that is a guess on my part. I saw a set of shimmy damper cylinders on E-bay of all places, and the photos seem to show clearance there, whereas on a classic hydarulic linear actuator there would be no clearance at all; just a tight seal against the rod surface.
By the way, the way this is solved in a reciprocating piston engine is that the piston heads have a rotational joint which attaches them to the connecting links. I don't think the shimmy damper has such a thing inside that little cylinder.