First, suppose $V$ is a vector space and $\Omega\subset V$ is a subset. Define a "shadow" of $\Omega$ as the image $\pi(\Omega)$, where $\pi:V\rightarrow W$ is a linear transformation. In this context, I am thinking of $\pi$ as an orthogonal projection, where $W$ is a subspace of $V$, although it is not required to make this identification.
Next, suppose $V$ is a vector space and $\Omega\subset V$ is a subset. Define a "permutation symmetry" of $\Omega$ as a permutation of $\Omega$ which extends to a linear operator on $V$. For example, suppose $V=\mathbb{R}^d$ and $\Omega\subset V$ has $n$ points. Then one may regard $\Omega$ as a $d\times n$ matrix and a permutation $\sigma\in\mathrm{Perm}(\Omega)$ is a permutation symmetry if there is a $d\times d$ matrix $M$ such that $M\Omega=\Omega P_\sigma$, where $P_\sigma$ is the permutation matrix corresponding to $\sigma$.
Next, suppose $V$ is a vector space and $\Omega\subset V$ is a subset. Call a linear transformation $\pi:V\rightarrow W$ "generic" with respect to $\Omega$ if the restriction of $\pi$ to $\Omega$ is a bijection. For simplicity, regard $\Omega$ as finite so that a projection $\pi$ is generic if $|\pi(\Omega)| = | \Omega|$.