Symmetry in machine construction means that the left and right sides, front and rear assemblies are built according to the same dimensions and positions. This sounds obvious, but it often isn't in evolved machine designs. If a machine has been gradually developed over years, asymmetries arise: an additional component here, a different screw position there.
Symmetry eliminates these asymmetries from the ground up. The result is assemblies that can be used symmetrically. Instead of a left and a right part, there is one part that can be used on both sides. The variety of variants decreases, and the number of parts in the bill of materials decreases.
Common parts reinforce this effect. Where previously two similar but not identical components were used, there is now one. This has direct effects on purchasing, warehousing, manufacturing, and assembly: fewer variants mean more favorable purchasing conditions, lower storage costs, simpler manufacturing processes, and fewer sources of error in assembly.