Symmetrisation inequlity for moments

Question

There is an inequality that I feel should be true but I am not sure how to prove it or if it is true. Let $X$ be a centered random variable and let $X^\prime$ be an independent copy. Is it true that $\|X\|_p\le 2^{-1/p} \|X-X^\prime\|_p$. This is definitively true if $p=2$ or if $X$ is a Normal or a recentered Bernouilli. But I am not sure how to prove it.