The link pivots on either a bushing or a ball joint at each attachment point.
In the attached photo of a 5-link live axle suspension, the different types of links can be seen.
These links work in tandem with the coil springs, dampers, and sway bar to control all six degrees of freedom of the axle.
The upper links (orange) and the lower links (yellow) work in tandem to control the pitch, yaw and the fore and aft movement (surge).
The panhard rod (green) controls the left and right movement (sway).