  We introduce a notion of the twist of an isometry of the hyperbolic plane.
This twist function is defined on the universal covering group of
orientation-preserving isometries of the hyperbolic plane, at each point in the
plane. We relate this function to a function defined by Milnor and generalised
by Wood. We deduce various properties of the twist function, and use it to give
new proofs of several well-known results, including the Milnor--Wood
inequality, using purely hyperbolic-geometric methods. Our methods express
inequalities in Milnor's function as equalities, with the deficiency from
equality given by an area in the hyperbolic plane. We find that the twist of
certain products found in surface group presentations is equal to the area of
certain hyperbolic polygons arising as their fundamental domains.
