  Given a triangulation of a point set in the plane, a \emph{flip} deletes an
edge $e$ whose removal leaves a convex quadrilateral, and replaces $e$ by the
opposite diagonal of the quadrilateral. It is well known that any triangulation
of a point set can be reconfigured to any other triangulation by some sequence
of flips. We explore this question in the setting where each edge of a
triangulation has a label, and a flip transfers the label of the removed edge
to the new edge.
  It is not true that every labelled triangulation of a point set can be
reconfigured to every other labelled triangulation via a sequence of flips. We
characterize when this is possible by proving the \emph{Orbit Conjecture} of
Bose, Lubiw, Pathak and Verdonschot which states that \emph{all} labels can be
simultaneously mapped to their destination if and only if \emph{each} label
individually can be mapped to its destination.
  Furthermore, we give a polynomial-time algorithm to find a sequence of flips
to reconfigure one labelled triangulation to another, if such a sequence
exists, and we prove an upper bound of $O(n^7)$ on the length of the flip
sequence.
  Our proof uses the topological result that the sets of pairwise non-crossing
edges on a planar point set form a simplicial complex that is homeomorphic to a
high-dimensional ball (this follows from a result of Orden and Santos; we give
a different proof based on a shelling argument). The dual cell complex of this
simplicial ball, called the \emph{flip complex}, has the usual flip graph as
its $1$-skeleton. We use properties of the $2$-skeleton of the flip complex to
prove the Orbit Conjecture.
