  We introduce the problem Synchronized Planarity. Roughly speaking, its input
is a loop-free multi-graph together with synchronization constraints that,
e.g., match pairs of vertices of equal degree by providing a bijection between
their edges. Synchronized Planarity then asks whether the graph admits a
crossing-free embedding into the plane such that the orders of edges around
synchronized vertices are consistent. We show, on the one hand, that
Synchronized Planarity can be solved in quadratic time, and, on the other hand,
that it serves as a powerful modeling language that lets us easily formulate
several constrained planarity problems as instances of Synchronized Planarity.
In particular, this lets us solve Clustered Planarity in quadratic time, where
the most efficient previously known algorithm has an upper bound of $O(n^{8})$.
