The Equational Meaning of Deeds

A deed is a closed normal form of the shape $\lambda x.xP_1\cdots P_n$ for some $n\geq0$. Let $M\mapsto D_M$ be the injective encoding from arbitrary normal forms to deeds described in the source. For arbitrary normal forms $M$ and $N$, what properties of the solutions $X$ of

$$MX =_{\beta\eta} NX$$

can be inferred from, or characterized by, the solutions of

$$D_MX =_{\beta\eta} D_NX?$$