A simple fact is that $P = NP \to P = coNP$, which follows from the observation that $P$ is closed under complement.
I am having trouble seeing that an analogous statement is true at higher levels of $PH$. For example, is it known that $NP = \Sigma_2 P$ implies $NP = \Pi_2 P$? If so, is there an easy proof? Would such a statement have any other interesting implications (For example $NP = coNP$)?
It seems somewhat likely to me that this is true, based on the observation that $NP = \Sigma_2 P$ means that $\exists \forall$ quantifier patterns can be replaced with $\exists$ quantifier patterns, and so all higher levels of $PH$ would at least collapse to $\Pi_2 P$.