There is an alternative strategy for cutting up the Bring sextic into pairs of pants that might compete with the decomposition above in some ways.
Of the $30$ loops of the second-shortest length $2\operatorname{arccosh}((11+5\sqrt{5})/4)\approx 4.796$, there are sets of six that are disjoint. Cutting along such a set of six breaks the Bring sextic up into three pieces, each of which has four loops of boundary --- as happened also in the decomposition above.
The Bring sextic has $10$ loops of the third-shortest length, which is $2\operatorname{arccosh}((26+10\sqrt{5})/4)\approx 6.368$. For each of our current three pieces, there are two of those $10$ third-shortest loops, either of which can cut up that piece into two pairs of pants. With three binary choices, we get eight decompositions into isometric pairs of pants. The 3-regular graph that gives the sewing of the cuffs is $K_{3,3}$ in four of those eight decompositions, but is the edge graph of a triangular prism in the other four. The twists are somewhat simpler in all eight of those decompositions, however --- which might make them more attractive than the decomposition above for some purposes: The twists along the six shorter loops are $1/4$, as for the three loops of that same length in the decomposition above, but the twists along the three longer loops are $0$.