  Given a drawing $D$ of a graph $G$, we define the crossing number between any
two cycles $C_{1}$ and $C_{2}$ in $D$ to be the number of crossings that
involve at least one edge from each of $C_1$ and $C_2$ except the crossings
between edges that are common to both cycles. We show that if the crossing
number between every two cycles in $G$ is even in a drawing of $G$ on the
plane, then there is a planar drawing of $G$. This result can be extended to
arbitrary surfaces. We also establish an equivalence between our result and a
fundamental result due to Cairns-Nikolayevsky and
Pelsmajer-Schaefer-\v{S}tefankovi\v{c}, about drawing graphs on surfaces, and
derive the Loebl-Masbaum theorem from it.
