  We discuss differences between genera of smooth and locally-flat
non-orientable surfaces in the 4-ball with boundary a given torus knot or
2-bridge knot. In particular, we establish that a result by Batson on the
smooth non-orientable 4-genus of torus knots does not hold in the locally-flat
category. We further show that certain families of torus knots are not the
boundary of an embedded M\"obius band in the 4-ball and other 4-manifolds.
  Our investigation of non-orientable surfaces with boundary a given torus knot
is motivated by our approach to unify the proof of the existence of inscribed
squares and of inscribed rectangles with aspect ratio $\sqrt3$ in Jordan curves
with a regularity condition. This generalizes a result by Hugelmeyer for smooth
Jordan curves.
