Let $A$ and $B$ be real (symmetric) and positive definite. It follows that $AB+BA$ is not necessarily positive definite (it can be indefinite, negative definite or positive definite).
But now suppose $A\geq B$. Can one always say $2A^2 \geq AB+BA$?
Since this ordering implies $2A^2-AB-BA\geq0$ and adding indefinite/neg-def/pos-def to a pos-def matrix may still be pos-def I am supposing the question is well-defined.