5. Let the function $f(x)$ be defined on $\mathbf{R}$, and consider the following three statements:
(1) The graph of a function satisfying $f(x+2)+f(2-x)=4$ is symmetric about the point $(2,2)$;
(2) The graph of a function satisfying $f(x+2)=f(2-x)$ is symmetric about the line $x=2$;
(3) The graphs of the functions $f(x-2)$ and $f(-x+2)$ in the same coordinate system are symmetric about the line $x=2$.
Among these, the number of true statements is ( ).
(A) 0
(B) 1
(C) 2
(D) 3
5. D.

Substitute $x-2$ for $x$ in $f(x+2)+f(2-x)=4$, we get $f(x)+f(4-x)=4$.

If the point $(x, y)$ is on the graph of $y=f(x)$, then $4-y=f(4-x)$, which means the point $(x, y)$'s symmetric point about $(2,2)$, $(4-x, 4-y)$, is also on the graph of $y=f(x)$. Conversely, this is also true, so proposition (1) is a true proposition.

Substitute $x-2$ for $x$ in $f(x+2)=f(2-x)$, we get $f(x)=f(4-x)$.

If the point $(x, y)$ is on the graph of $y=f(x)$, then $y=f(4-x)$, which means the point $(x, y)$'s symmetric point about $x=2$, $(4-x, y)$, is also on the graph of $y=f(x)$, so proposition (2) is a true proposition.

Since proposition (2) is a true proposition, it is not difficult to deduce that proposition (3) is also a true proposition.
Therefore, all three propositions are true propositions.