  Using quaternionic Feix--Kaledin construction we provide a local
classification of quaternion-K\"ahler metrics with a rotating $S^1$-symmetry
with the fixed point set submanifold $S$ of maximal possible dimension. For any
K\"ahler manifold $S$ equipped with a line bundle with a unitary connection of
curvature proportional to the K\"ahler form we explicitly construct a
holomorphic contact distribution on the twistor space obtained by the
quaternionic Feix-Kaledin construction from this data. Conversely, we show that
quaternion-K\"ahler metrics with a rotating $S^1$-symmetry induce on the fixed
point set of maximal dimension a K\"ahler metric together with a unitary
connection on a holomorphic line bundle with curvature proportional to the
K\"ahler form and the two constructions are inverse to each other. Moreover, we
study the case when $S$ is compact, showing that in this case the
quaternion-K\"ahler geometry is determined by the K\"ahler metric on the fixed
point set (of maximal possible dimension) and by the contact line bundle.
Finally, we relate the results to the c-map construction showing that the
family of quaternion-K\"ahler manifolds obtained from a fixed K\"ahler metric
on $S$ by varying the line bundle and the hyperk\"ahler manifold obtained by
hyperk\"ahler Feix--Kaledin construction form $S$ are related by
hyperk\"ahler/quaternion-K\"ahler correspondence.
