FullSimplify a expression with a even function“`Df[x] == Df[-x]“`?

I want to simplify a expression with a even functionDf[x] == Df[-x] like:

FullSimplify[-(1/2) Df[x - x4] J[x] - 1/2 Df[-x + x4] J[x], Assumptions -> {Df[x] == Df[-x]}]

This code can not give the desired result like -Df[-x + x4] J[x].

And the strange thing here is that if I replace x4 with x2, it works well.

FullSimplify[-(1/2) Df[x - x2] J[x] - 1/2 Df[-x + x2] J[x], Assumptions -> {Df[x] == Df[-x]}] will give -Df[-x + x2] J[x].

So any solutions here for x4 or arbitrary symbol?