Verification: Does symmetry of $f: \mathbb R \to \mathbb R$ imply symmetry of every derivative?

Question

Let $f: \mathbb R \to \mathbb R$ be differentiable and in $\mathbb R^2$ (point or axially) symmetric, that means:

$$\exists a \in \mathbb R \, \forall x \in \mathbb R: f(a-x)+f(a+x)=2f(a)$$

Does that imply that every derivative is point or axially symmetric? Is there a counter example?