BALTIC-WAY 1998 Problem 6

Let $P$ be a polynomial of degree 6 and let $a, b$ be real numbers such that $0<a<b$. Suppose that $P(a)=P(-a), P(b)=P(-b)$ and $P^{\prime}(0)=0$. Prove that $P(x)=P(-x)$ for all real $x$.