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We investigate sums $K(\vec{x})$ and $L(\vec{x})$ of pairs of (suitablynormalized) Saalsch\"utzian ${}_4F_3(1)$ hypergeometric series, and develop atheory of relations among these $K$ and $L$ functions.  The function $L(\vec{x})$ has been studied extensively in the literature, andhas been shown to satisfy a number of two-term and three-term relations withrespect to the variable $\vec{x}$. More recent works have framed theserelations in terms of Coxeter group actions on $\vec{x}$, and have developed asimilar theory of two-term and three-term relations for $K(\vec{x})$.  In this article, we derive "mixed" three-term relations, wherein any one ofthe $L$ (respectively, $K$) functions arising in the above context may beexpressed as a linear combination of two of the above $K$ (respectively, $L$)functions. We show that, under the appropriate Coxeter group action, theresulting set of three-term relations (mixed and otherwise) among $K$ and $L$functions partitions into eighteen orbits. We provide an explicit example of arelation from each orbit.  We further classify the eighteen orbits into five types, with each typeuniquely determined by the distances (under a certain natural metric) betweenthe $K$ and $L$ functions in the relation. We show that the type of a relationdictates the complexity (in terms of both number of summands and number offactors in each summand) of the coefficients of the $K$ and $L$ functionstherein.

Relations among complementary and supplementary pairings of Saalschutzian 4F3(1) series