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. All nine of the twists are zero in all eight of those decompositions, however --- which might make them more attractive than the decomposition above for some purposes.

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 $0$.