Is the total strain energy vs. tensile and compressive strain symmetric?
It is not because strain measures deformation, not stress or force of any sort. Consider an elastic solid - if you compress it the modulus changes very fast as it reaches the limits of elastic compression, whereas if you stretch it you might find quite a small change right up to the point that it broke. However if you measure strain energy versus tensile and compressive stress then I would expect much greater symmetry.
It is symmetric if the strains are small, i.e. for the linear or Hook's law elasticity. For an ideal (Hooks law) spring the energy is (1/2)kx^2, where positive x present elongation and negative ones present contraction. But for larger strains, i.e. for non-linear elasticity, this symmetry is lost. The energy of non-ideal spring is (1/2)kx^2+px^3+ terms of higher order. The meaning of "small" and " large" depends on the material and the desired precision.
For the elastic Hookian and hyperelastic solıds where one has linear connection between the stress and strain tensors regardless one has finite deformation or not the strain energy density which is defined by 1/2 Sigma: Epsilon where one has double tensor inner product and the sign of the deformation whether its trace is positive (compression) or negative (triaxial tension or expansion) doesn't matter.
If we refer to a simple model of chemical bonding, the effects of compression and tension are not symmetrical. Only this model is not verified experimentally since the computed breaking stress is much greater (100 times or more) than the measured stress. To go further, it is necessary to introduce the energy localization on the defects. A microscopic approach predicts that defects do not have a symmetrical effect depending on whether they are acceptors or electron donors. Spectroscopic studies show that applied stresses and stresses around defects increase the vibration frequencies of atoms. It would appear that positive charges produce larger deformations than those due to negative charges
You will find additional information in the following reference: Electron-trapping and energy localization in insulating materials. Technological impact of space charge electron-beam characterization
AIP Advances 8, 095228 (2018); https://doi.org/10.1063/1.5047673
For the elastic region yes it is fully symmetric! That means no hysteresis loss for the cyclic loading as well.
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