ISL 2014 A5

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