The question is: for a given positive integer $n$, do there always exist positive integers $x,y$ such that $\phi(x)+\phi(y)=2n$?
The case $n=1$ is good using $x=y=1$. Assume $n>1$. If we invoke Goldbach's conjecture, then yes the claim is true. Let $p$ and $q$ be primes st $p+q=2(n+1)$. Then $$\phi(p)+\phi(q)=(p-1)+(q-1)=p+q-2=2n.$$ But do we know anything about this equation without using the conjecture? Thanks.
Maybe we can use what Keith Backman suggested that even nontotients are rare. Let $n$ be an even integer $>0$. So the set $S_n$ of even nontotients $\leq n$ is a "large subset of an arithmetic progression" and we might be able to use some sum sets inequalities to show that $S_n+S_n$ contains all even integers in $4,6,...,n$. I still haven't found any useful results about sum sets to use.