  We show how to recover a discrete twist over an ample Hausdorff groupoid from
a pair consisting of an algebra and what we call a quasi-Cartan subalgebra. We
identify precisely which twists arise in this way (namely, those that satisfy
the local bisection hypothesis), and we prove that the assignment of twisted
Steinberg algebras to such twists and our construction of a twist from a
quasi-Cartan pair are mutually inverse. We identify the algebraic pairs that
correspond to effective groupoids and to principal groupoids. We also indicate
the scope of our results by identifying large classes of twists for which the
local bisection hypothesis holds automatically.
