Školjka

Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has $$ |y^2 - P(x)| \leq 2 |x| \quad \text{if and only if} \quad |x^2 - P(y)| \leq 2 |y| \text{.} $$ Determine all possible values of $P(0)$. \begin{flushright}\emph{(Belgium)}\end{flushright}