A lattice is a discrete (we may assume full-rank) subgroup of $\mathbb{R}^n$, often written as the image of $\mathbb{Z}^n$ under a particular matrix $\mathbf{B}\in\mathbb{R}^{n\times n}$ (a basis of the lattice).
Associated with any lattice $L$ are two different objects:
The Voronoi cell of $L$: $\mathcal{V}_L = \{x\in\mathbb{R}^n \mid \forall \ell\in L\setminus \{0\}, \|x\|_2^2 < \|x - \ell\|_2^2\}$
The Dual lattice of $L$: $L^\vee = \{x\in\mathbb{R}^n \mid \forall \ell\in L, \langle \ell, x\rangle \in\mathbb{Z}\}$
As the dual lattice is itself a lattice, it has its own Voronoi cell $\mathcal{V}_{L^\vee}$. There are known relationships between $\mathcal{V}_L$ and $\mathcal{V}_{L^\vee}$. In particular, it is known that:
$$\mathsf{vol}(\mathcal{V}_L)\mathsf{vol}(\mathcal{V}_{L^\vee}) = 1$$
There are other somewhat deeper relationships known between various numerical quantities related to the Voronoi cell of the primal and dual lattices such as transference theorems. One famous one is that: $$1 \leq \lambda_1(L)\rho(L^\vee) \leq n$$ (I may be missing some constant factors) where $\lambda_1(L)$ is (a scaling of) the inradius of $\mathcal{V}_L$ and $\rho(L^\vee)$ is the circumradius of $\mathcal{V}_{L^\vee}$. My question is:
Are "non-numeric" relationships known between $\mathcal{V}_L$ and $\mathcal{V}_{L^\vee}$?
By this, I mean ways of using knowledge of $\mathcal{V}_L$ (as a polyhedron) to determine attributes of $\mathcal{V}_{L^\vee}$ (as a polyhedron) that aren't simply relationships between $\mathbb{R}$-valued functions of the polyhedra. For example, if one could use knowledge of the location of the vertices of $\mathcal{V}_L$ to determine the vertices of $\mathcal{V}_{L^\vee}$.