When a compact Riemann surface of genus $g$ is cut up along $3g-3$ disjoint geodesic loops into $2g-2$ pairs of pants, the result is often described by giving Fenchel-Nielsen coordinates: one length and one twist for each of the $3g-3$ loops along which the cutting happens. The lengths of the loops are uniquely determined, but the values for the twists depend upon some extra structure added to the surface called a marking. There are infinitely many possible markings and no clear way to choose a "best" marking. You can learn more about all this from various sources, including
- the book The Real Analytic Theory of Teichmüller Space, by William Abikoff, (recommended by Moishe Kohan in a comment to an earlier draft of this question);
- Chapter 10 of the book A Primer on Mapping Class Groups, by Benson Farb and Dan Margalit;
- and the paper Teichmüller space and Fenchel-Nielsen coordinates, by Nathan Lopez.
The twist along a loop $L$ is determined, by the marking, from four points on $L$ that form two antipodal pairs. There are markings that will report, as the twist along $L$, the fraction of the length of $L$ that is involved in traveling from either point of one pair to either point of the other pair, in either direction, including possibly wrapping around all of $L$ some number of times in the process. To say that another way, let $t$ be the smallest fraction of $L$ that separates a point of one pair from a point of the other pair; so $0\le t\le \frac{1}{4}$. There are then markings that will report, as the twist, any number that differs by an integer from either $t$, $1-t$, $\frac{1}{2}+t$, or $\frac{1}{2}-t$.
When people describe some explicit decomposition of some explicit Riemann surface into pairs of pants, they often give the Fenchel-Nielsen coordinates without specifying a marking. They must have chosen a marking that struck them as the most natural or the simplest in some sense, used that marking to determine the twist, but then left it implicit.
One simple rule would simply report $t$ as the twist. But there are lots of examples in which a reported twist lies, not in $\bigl[0\mathop{..}\frac{1}{4}\bigr]$, but instead in $\bigl(\frac{1}{4}\mathop{..}\frac{1}{2}\bigr]$. I originally asked whether someone could clarify for me why these larger values are sometimes reported.
I now have a plausible theory that explains all of the large twist values that I've seen reported. I will describe that theory as my suggested answer to my own question.
Let $L$ be one of the loops along which we cut, and let $P$ and $Q$ be the two pair of pants on the two sides of $L$. (It can happen that $P=Q$.) There are shortest geodesic arcs in $P$ that connect $L$ to each of the other two boundary loops of $P$. Let's call those other two loops $A$ and $B$. (It can happen that some two of the three loops $L$, $A$, and $B$ coincide; but they can't all three coincide.) Let $L_{PA}$ be the point along $L$ where the shortest arc to $A$ starts out, and similarly for $L_{PB}$. The points $L_{PA}$ and $L_{PB}$ are always antipodal along $L$ (even when some two of $L$, $A$, and $B$ coincide).
In a similar way, let's denote the other two loops that bound the pair of pants $Q$ by $C$ and $D$. And let $L_{QC}$ and $L_{QD}$ be the start points of the shortest geodesic arcs in $Q$ that lead from $L$ to $C$ and to $D$. The points $L_{QC}$ and $L_{QD}$ are also antipodal along $L$. The marking determines the twist from the two pairs of points $(L_{PA},L_{PB})$ and $(L_{QC},L_{QD})$.
We always have $\overline{L_{PA}L_{QC}}=\overline{L_{PB}L_{QD}}$ and $ \overline{L_{PA}L_{QD}}=\overline{L_{PB}L_{QC}}$. The first of those corresponds to matching up the pairs $(L_{PA},L_{PB})$ and $(L_{QC},L_{QD})$ by matching $A$ to $C$ and $B$ to $D$; the second matches $A$ to $D$ and $B$ to $C$. If we measure distances as fractions of the overall length of $L$, those two repeated distances sum to $\frac{1}{2}$. In the notation of the question, the smaller of those two distances is $t$ and the larger is $\frac{1}{2}-t$.
If $A$, $B$, $C$, and $D$ are four distinct loops of four distinct lengths, I see no reason not to report the smaller value, $t$, as the twist. There are various special cases, though, in which one of the two matchings between the sets $\{A,B\}$ and $\{C,D\}$ might seem more natural --- and people would then probably report the twist associated with the more natural matching.
One obvious source of such special cases is when the sets $\{A,B\}$ and $\{C,D\}$ overlap. For example, if $A=C$, then we would want to pair $A$ with $C$. Luckily, whenever some equality between loops votes for one of the matchings, there won't be any votes for the other matching. For example, if $A=C$, we may or may not also have $B=D$, which would be a second vote for the same matching; but we definitely won't have either $A=D$ or $B=C$, which would be votes for the other matching, since every loop has only two sides on which pairs of pants can lie.
Even if the four loops are distinct, it might happen that $A$ and $B$ have two different lengths while $C$ and $D$ have those same two different lengths. It would then make sense to pair up $A$ with whichever of $C$ and $D$ has the same length.
Whatever. My best current guess is that, whenever a twist greater than $1/4$ is reported without describing any marking, that happens because the author saw some reason to prefer one matching over the other.