The key trick here is to realize that the relation is of the form :
{ordered pair, ordered pair} and not simply ordered pair.
Ok, so for reflexive
$\forall_{a,b}\, if((a,b),(a,b)) \in \mathrel{R} \rightarrow \text{reflexive}$
$((a,b),(a,b)) \in \mathrel{R} \;\leftrightarrow(a-b=b-a) $ (not possible for any postive integers b and a)
but that is a contradiction hence it is not reflexive.
now, for symmetric
$((a,b),(c,d))\in \mathrel{R}\rightarrow((c,d),(a,b))\in \mathrel{R}$
$((a,b),(c,d))\in\mathrel{R} \rightarrow(a-d=b-c)$
$((c,d),(a,b))\in\mathrel{R}$
$\because (c-b=d-a)\leftrightarrow (d-a=c-b)\leftrightarrow(-(a-d)=-(b-c))\leftrightarrow(a-d=b-c)$
So, it is symmetric.
Hence, C is the correct option.