Find the ordered pair $(x,y)$ if

\begin{align*}
x+y&=(3-x)+(3-y),\\
x-y &=(x-2)+(y-2).
\end{align*}
Adding the equations, we get  $$2x=2\Rightarrow x=1.$$ Substituting this into the first equation, we get $$1+y=5-y\Rightarrow y=2.$$ Thus the ordered pair is $\boxed{(1,2)}$.