  We introduce a common generalization of the strong Hanani-Tutte theorem and
the weak Hanani-Tutte theorem: if a graph $G$ has a drawing $D$ in the plane
where every pair of independent edges crosses an even number of times, then $G$
has a planar drawing preserving the rotation of each vertex whose incident
edges cross each other evenly in $D$. The theorem is implicit in the proof of
the strong Hanani-Tutte theorem by Pelsmajer, Schaefer and \v{S}tefankovi\v{c}.
We give a new, somewhat simpler proof.
