Symmetric property of DDT of inverse function
The proof for DDT is shown as following:

given:
$$y=F(x) ;\\ x=F^{-1}(y) \&gt; \&gt; (1) $$
The forward DDT is 
$$F(x) + F(x+\alpha) = \beta \&gt;\&gt; (2)$$
substituting (1) into (2)
$$F(x) + F(F^{-1}(y)+\alpha) = \beta  $$

$$y+ \beta +F(F^{-1}(y)+\alpha) =0$$

re-arranging and applying inverse function
$$F^{-1}(y+ \beta) +F^{-1}((F(F^{-1}(y)+\alpha))=0 $$

$$F^{-1}(y+ \beta) +F^{-1}(y) = \alpha$$

Therefore , 

$$DDT_{F^{-1}}(\beta, \alpha) = \delta_{F^{-1}}(\beta, \alpha) = |\{ x \in \mathbb{F}_2^n | F^{-1}(x) + F^{-1}(x+\beta) = \alpha  \}|$$

The same concept applied to LAT proof.