In this paper, we introduce a symmetry geometry for all those mathematical structures which can be characterized by means of a generalization (which we call pairing) of a finite rectangular table. In more detail, let $\unicode[STIX]{x1D6FA}$ be a given set. A pairing$\mathfrak{P}$ on $\unicode[STIX]{x1D6FA}$ is a triple $\mathfrak{P}:=(U,F,\unicode[STIX]{x1D6EC})$, where $U$ and $\unicode[STIX]{x1D6EC}$ are nonempty sets and $F:U\times \unicode[STIX]{x1D6FA}\rightarrow \unicode[STIX]{x1D6EC}$ is a map having domain $U\times \unicode[STIX]{x1D6FA}$ and codomain $\unicode[STIX]{x1D6EC}$. Through this notion, we introduce a local symmetry relation on $U$ and a global symmetry relation on the power set ${\mathcal{P}}(\unicode[STIX]{x1D6FA})$. Based on these two relations, we establish the basic properties of our symmetry geometry induced by $\mathfrak{P}$. The basic tool of our study is a closure operator $M_{\mathfrak{P}}$, by means of which (in the finite case) we can represent any closure operator. We relate the study of such a closure operator to several types of others set operators and set systems which refine the notion of an abstract simplicial complex.