  In this paper, we use combinatorial group theory and a limiting process to
connect various types of hypergeometric series, and of relations among such
series.
  We begin with a set $S$ of 56 distinct translates of a certain function $M$,
which takes the form of a Barnes integral, and is expressible as a sum of two
very-well-poised $_9F_8$ hypergeometric series of unit argument. We consider a
known, transitive action of the Coxeter group $W(E_7)$ on this set. We show
that, by removing from $W(E_7)$ a particular generator, we obtain a subgroup
that is isomorphic to $W(D_6)$, and that acts intransitively on $S$,
partitioning it into three orbits, of sizes 32, 12, and 12 respectively.
  Taking certain limits of the $M$ functions in the first orbit yields a set of
32 $J$ functions, each of which is a sum of two Saalsch\"utzian $_4F_3$
hypergeometric series of unit argument. The original action of $W(D_6)$ on the
$M$ functions in this orbit is then seen to correspond to a known action of
this group on this set of $J$ functions.
  In a similar way, the image of each of the size-12 orbits, under a similar
limiting process, is a set of 12 $L$ functions that have been investigated in
earlier works. In fact, these two image sets are the same.
  The limiting process is seen to preserve distance, except on pairs consisting
of one $M$ function from each size-12 orbit.
  Finally, each known three-term relation among the $J$ and $L$ functions is
seen to be obtainable as a limit of a known three-term relation among the $M$
functions.
