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.

A sporting league places every team into one of several divisions of equal size, and runs a round robin tournament for each division. Some teams are paired with another team, not necessarily in the same division, to share facilities. It is shown that however many teams are paired and whatever the pairings, it is always possible to schedule the fixtures in the minimum time, so that no two paired teams have home matches simultaneously.