$NP\cap coNP$ is low for itself: $NP^{NP\cap coNP}\cap coNP^{NP\cap coNP}=NP\cap coNP$.
Going up to exponential realm, we are considering similarities of $NP\cap coNP$ to its exponential version $NEXP\cap coNEXP$.
By padding, we already have $$NEXP^{NP\cap coNP}\cap coNEXP^{NP\cap coNP}=NEXP\cap coNEXP$$
Two questions:
Though it is unlikely, I do not know for sure whether $NEXP\cap coNEXP$ is low for itself: $$NEXP^{NEXP\cap coNEXP}\cap coNEXP^{NEXP\cap coNEXP}=NEXP\cap coNEXP$$
Being more moderate in using oracle machine, we may ask whether $$P^{NEXP\cap coNEXP}=NEXP\cap coNEXP$$