  Various structural properties are developed for non-orientable surfaces in
link spaces. The M\"obius band tree is described to represent genus growth of
one-sided surfaces in solid tori. The structure of the Tree allows various
insights into the change of genus under boundary slope, which are not possible
using the existing continued fractions algorithm. A restriction under which
geometrically incompressible, boundary compressible one-sided surfaces have a
unique boundary incompressible form away from the boundary is established.
