Taylor series of upx function

[Kouznetsov]

Thank you.

What did you find about simmetricity?

If y=J[x], then, the symmetry with respect to line x=-y implies that
-x=J[-y], and then
-x=J[-J[x]].
Please, plot J[-J[x]]+1 versus x; if symmetry, then, it should be zero.
You may just add the line

PJ = Plot[{J[-J[z]] + z}, {z, -1.999, 1.9}, PlotRange -> All];

to the coide, and get the graphic below:
   

The deviation is small, of order of a percent. Could you confirm this result?