It is not known whether complete languages exist for $NP\cap CoNP$, and Sipser has shown that there are relativizations so that $NP\cap CoNP$ has no $\leq ^{P}{m}$-complete languages. In this paper we show that $NP\cap CoNP$ has $\leq ^{P}{m}$-complete languages if and only if it has $\leq ^{P}{T}$-complete languages. Furthermore, we show that if a complete language $L{0}$ exists for $NP\cap CoNP$ and $NP\cap CoNP \neq NP$ then the reduction of $L(N_{i}) \in NP\cap CoNP$ cannot be effectively computed from $N_{i}$. We extend the relativization results by exhibiting an oracle $E$ such that $P^{E} \neq NP^{E} \cap CoNP^{E} \neq NP^{E}$ and for which there exist complete languages in the intersection. For this oracle the reduction to a complete language can be effectively computed from complementary pairs of machines $(N_{i}, N_{j})$ such that $L(N_{i})= \overline{L(N_{j})}$. On the other hand, there also exist oracles $F$ such that $P^{F} \neq NP^{F} \cap CoNP^{F} \neq NP^{F}$ for which the intersection has complete languages, but the reductions to the complete language cannot be effectively computable from the complementary pairs of machines. In this case, the reductions can be computed from $(N_{i}, N_{j}$, Proof that $L(N_{i})= \overline{L(N_{j})}).$