Let $i\dashv r\dashv i:\mathcal{B}\hookrightarrow \mathcal{A}$ be a bireflective subcategory. $r$ is necessarily left exact hence this a special type of essential localization called a 'quintessential localization' in Johnstone's 1996 TAC paper (see https://ncatlab.org/nlab/show/quality+type for references).

An *example* of a quintessential localization is the inclusion of $Set$ into the category with objects $(X,e:X\to X)$ where $X$ is a set and $e$ is an idempotent map $e^2=e$ and equivariant maps $f:(X_1,e_1)\to (X_2,e_2)$ with $f\cdot e_1=e_2\cdot f$. The inclusion $i$ maps a set $X$ to $(X,id_X)$ whereas the reflection $r$ maps $(X,e)$ to the set of fixed points $\{x|e(x)=x\}$. In this situation $r$ is left as well as right adjoint to $i$.

In their 1989 paper 'On the complete lattice of essential localizations' Kelly and Lawvere give the following characterization of essential localizations (see also Bedos&Quigg TAC 2011 no.20):

$\mathbf{Theorem\mbox{ }3.2}\quad$
A set $E$ of morphisms of a finitely bi-complete category $\mathcal{A}$ is of the form $\mathcal{B}^\perp$ for some essential localization $l\dashv r\dashv i:\mathcal{B}\hookrightarrow \mathcal{A}$ iff there exist factorization systems $(E,M)$ and $(N,E)$ on $\mathcal{A}$.

[Initially I then claimed here that it suffices to replace N by M in this proposition to get a specialization to quintessential localizations but now I think that this makes the situation too symmetric because then $(E,M)$ and $(M,E)$ yield localizations which either are the same in which case $E$ and $M$ coincide or there are in fact two different localizations. So after all, I have to retract the answer to the second question. I've decided to let stay the rest since it provides at least some useful information although in this form more suitable for a comment. My apologies!]

In theorem 2.4 Kelly-Lawvere show that essential localizations correspond to *associated pairs*.

$\mathbf{Def.}\quad$ A pair $(\mathcal{B},\mathcal{C})$ of subcategories of a category $\mathcal{A}$ is called an *associated pair* if $\mathcal{B}$ is reflective and $\mathcal{C}$ is coreflective and furthermore $\mathcal{B}^\perp=\mathcal{C}^\top$.

The properties given there specalize to quintessential subcategories $\mathcal{B}$ in the form that they correspond to associated pairs $(\mathcal{B},\mathcal{B})$ hence $\mathcal{B}^\perp=\mathcal{B}^\top$ and, furthermore, $\mathcal{B}^{\perp\top}=\mathcal{B}=\mathcal{B}^{\top\perp}$.

Let $(E,M)$ be the factorization system corresponding to the quintessential subcategory $\mathcal{B}$ by thm.3.4. Then from $E=\mathcal{B}^\perp$ and the above we get the following necessary condition on $E$:
$$E^\perp=E^\top\quad.$$