  We use unoriented versions of instanton and knot Floer homology to prove
inequalities involving the Euler characteristic and the number of local maxima
appearing in unorientable cobordisms, which mirror results of a recent paper by
Juhasz, Miller, and Zemke concerning orientable cobordisms. Most of the
subtlety in our argument lies in the fact that maps for non-orientable
cobordisms require more complicated decorations than their orientable
counterparts. We introduce unoriented versions of the band unknotting number
and the refined cobordism distance and apply our results to give bounds on
these based on the torsion orders of the Floer homologies. Finally, we show
that the difference between the unoriented refined cobordism distance of a knot
$K$ from the unknot and the non-orientable slice genus of $K$ can be
arbitrarily large.
