Symmetry of action stiffness affects the direction that the barrel moves under recoil. If the action is perfectly symmetric side-to-side, then the barrel tends to vibrate only in the vertical plane. If not, the recoil force causes more deflection on the weak side, causing the barrel to rotate toward the weak side. Harold Vaughn has a section in his book on this.
It is useful to think about the relative stiffness of the action compared to the breech end of the barrel. The barrel is 1.25" OD and about 0.48" ID for a 30BR chamber. The action is 1.35" OD and 1" ID give or take a bit for lug races, etc. Comparing the moment of inertia of these sections, the action is not much stiffer than the barrel, which would be the ideal way to design the system. Then we cut a big loading port in one side and make the stiffness asymmetric. It is amazing rifles with such asymmetric actions shoot as well as they do. A possible explanation is that the bullet is launched in the same direction so long as the rifle responds exactly the same, and if the bullet exit time is the same. But given that bullet exit time varies, there is a benefit in having symmetric action stiffness. The effect may often be lost in the wash of other sources of dispersion, but in the effort to build increasingly accurate rifles, it is something to address.
Keith