What is symmetric?Please explain geometrically and in terms of displacements. I have encountered this while looking up for engineering strain and true strain values have totally different compression and expansion parameters for their respective strains.
My guess would be that talking about symmetry in the context of material science, the meaning of "symmetry" is probably describing direction-dependant properties of a material.
Steel for example is an isotropic material (excluding drawn/stretched steel), meaning that its properties are not dependent on the direction in which you measure them.
Note, this does not mean that compressive and tensile value must be the same. It basically means that if you have for material sample, you will measure the same properties, whether you measure them in $x$, $y$ or $z$ direction.
On the other hand, wood, reinforced concrete and other composite materials are usually not isotropic. Wood for example has a much higher strength when loaded in axial direction, compared to a radial or tangential load.
This makes wood an material with orthotropic symmetry, because its properties mainly vary along the three orthogonal directions (ax/rad/tang).
There's a formal explanation on the whole subject on Wikipedia, but it requires some understanding of linear algebra.