  We investigate the properties of knots in S^3 which bound Klein bottles, such
that a pushoff of the knot has zero linking number with the knot, i.e. has zero
framing. This is motivated by the many results in the literature regarding
slice knots of genus one, for example, the existence of homologically essential
zero self-linking simple closed curves on genus one Seifert surfaces for
algebraically slice knots. Given a knot K bounding a Klein bottle F with zero
framing, we show that J, the core of the orientation-preserving band in any
disk-band form of F, has zero self-linking. We prove that such a K is slice in
a Z[1/2]-homology 4-ball if and only if J is as well, a stronger result than
what is currently known for genus one slice knots. As an application, we prove
that given knots K and J and any odd integer p, the (2, p) cables of K and J
are Z[1/2]-concordant if and only if K and J are Z[1/2]-concordant. In
particular, if the (2,1)-cable of a knot K is slice, K is slice in a
Z[1/2]-homology ball.
