  Given a nonorientable, locally flatly embedded surface in the $4$-sphere of
nonorientable genus $h$, Massey showed that the normal Euler number lies in
$\lbrace -2h,-2h+4,\ldots,2h-4,2h \rbrace$. We prove that every such surface
with knot group of order two is topologically unknotted, provided that the
normal Euler number is not one of the extremal values in Massey's range. When
$h$ is $1$, $2$, or $3$, we prove the same holds even with extremal normal
Euler number. We also study nonorientable embedded surfaces in the 4-ball with
boundary a knot $K$ in the 3-sphere, again where the surface complement has
fundamental group of order two and nonorientable genus $h$. We prove that any
two such surfaces with the same normal Euler number become topologically
isotopic, rel. boundary, after adding a single tube to each. If the determinant
of $K$ is trivial, we show that any two such surfaces are isotopic, rel.
boundary, again provided that they have non-extremal normal Euler number, or
that $h$ is $1$, $2$, or $3$.
