In biological systems, the joints are actuated antagonistically by muscles that can be moved coherently to achieve the desired displacement and coactivated with appropriate forces to vary joint stiffness. Inspired by this, there is an interest in developing bio-inspired robots suitable for low- and high-stiffness tasks. Mechanisms actuated by antagonist cables can be a reasonable approximation of biological joints. A study on the anti-parallelogram mechanism showed that the antagonistic forces (> 0) positively influence its stiffness, similar to the biological joints. This work investigates more general symmetric four-bar mechanisms with crossed/non-crossed limbs and top and base bars of unequal lengths for this property. Firstly, the cables are attached between the two unconnected pivot pairs in the four-bar mechanism, and their limits of movement are presented. Inside these limits, we show that the cable forces have a positive (resp. negative) influence on the stiffness of the mechanism when its limbs are crossed (resp. non-crossed). These results are validated experimentally in all cases. Subsequently, we consider alternate cable attachments for the mechanisms with non-crossed limbs to achieve coactivation. Examples show that coactivation is possible in these mechanisms but comes at the cost of a diminished range of movement. Among all the four-bar mechanisms considered, the anti-parallelogram mechanism offers the largest orientation range of (−π, π) for the top bar w.r.t. its base while providing coactivation and is thus the best choice.