Contained in-between each level of the polynomial hierarchy are various complexity classes, including $\Delta_i^{\text{P}}$, $\text{DP}$, $\text{BH}_k$, and $\Sigma_i^\text{P} \cap \Pi_i^\text{P}$. For lack of better terminology, I will refer to these and any others as intermediate classes between levels $i$ and $i+1$ in the polynomial hierarchy. For the purposes of this question, assume they are the classes contained in $\Sigma_{i+1}^\text{P} \cap \Pi_{i+1}^\text{P}$ but contain $\Sigma_i^\text{P}$ and/or $\Pi_i^\text{P}$. We want to avoid including $\Sigma_{i+1}^\text{P} \cap \Pi_{i+1}^\text{P}$, if possible, as it is trivially equivalent to $\text{PH}$ if it collapses to the ${i+1}^{th}$ level.
In addition, define the following:
$\text{DP}_i = \left \{ L \cap L' : L \in \Sigma_i^\text{P} \text{ and } L' \in \Pi_i^{\text{P}} \right \}$
The above is a generalization of the class $\text{DP}$ (also written $\text{D}^\text{P}$). In this definition, $\text{DP}$ is equivalent to $\text{DP}_1$. It is considered in another cstheory.se question. It is easy to see that $\text{DP}_i \subseteq \Delta_{i+1}^{\text{P}}$ and contains both $\Sigma_i^\text{P}$ and $\Pi_i^\text{P}$.
Reference Diagram:
Question:
Suppose that the polynomial hierachy collapses to the ${i+1}^{th}$ level, but does not collapse to the $i^{th}$ level. That is, $\Sigma_{i+1}^\text{P}=\Pi_{i+1}^\text{P}$ and $\Sigma_{i}^\text{P}\neq\Pi_{i}^\text{P}$.
Can we say anything more about the relationships between these intermediate classes themselves and others in any level below $i+1$? Is there a schema for a collection of complexity classes where, for every collection, the classes are equivalent if and only if the $\text{PH}$ collapses exactly to an arbitrarily chosen level?
Just as a followup, suppose that the hierarchy collapsed to any particular one of these intermediate classes (such as $\Delta_{i+1}^{\text{P}}$). Depending on the class selected, do we know if this collapse must continue to extend downwards, perhaps even to the $i^{th}$ level?
The above question was partially explored and answered in a paper by Hemaspaandra et. al:
A Downward Collapse within the Polynomial Hierarchy
Does someone happen to know of additional examples not mentioned in this paper or have further intuition as to what needs to happen in order for a class to accomplish this?