  Semitoric integrable systems were symplectically classified by Pelayo and Vu
Ngoc in 2009-2011 in terms of five invariants. Four of these invariants were
already well-understood prior to the classification, but the fifth invariant,
the so-called twisting index invariant, came as a surprise. Intuitively, the
twisting index encodes how the structure in a neighborhood of a focus-focus
fiber compares to the large-scale structure of the semitoric system and it was
originally defined by comparing certain momentum maps.
  In the first half of the present paper, we produce several new formulations
of the twisting index which give rise to dynamical, geometric, and topological
interpretations. More specifically, we describe it in terms of differences of
action variables, Taylor series, and homology cycles. In the second half of the
paper, we compute the twisting index invariant of a specific family of systems
with two focus-focus singular points (the so-called generalized coupled angular
momenta), which is the first time that the twisting index has been computed for
a system with more than one focus-focus point. Moreover, we also compute the
terms of the Taylor series invariant up to second order. Since the other
invariants of this family were already computed, this becomes the third family
of semitoric systems for which all invariants are known, after the coupled spin
oscillators and the coupled angular momenta.
