The function $f(x)$ is defined for all real numbers.

Consider the following three conditions, where $a$ is a real constant:

I $f(a-x)=f(a+x)$ for all real $x$.

II $f(2a-x)=f(x)$ for all real $x$.

III $f(a-x)=f(x)$ for all real $x$.

Which of these conditions is/are necessary and sufficient for the graph of $y=f(x)$ to have reflection symmetry in the line $x=a$?