  In this work, N\'ed\'elec elements on locally refined meshes with hanging
nodes are considered. A crucial aspect is the orientation of the hanging edges
and faces. For non-orientable meshes, no solution or implementation has been
available to date. The problem statement and corresponding algorithms are
described in great detail. As a model problem, the time-harmonic Maxwell's
equations are adopted because N\'ed\'elec elements constitute their natural
discretization. The algorithms and implementation are demonstrated through two
numerical examples on different uniformly and adaptively refined meshes. The
implementation is performed within the finite element library deal.II.
