  We investigate the nonorientable 4-genus $\gamma_4$ of a special family of
2-bridge knots, the twist knots and double twist knots $C(m,n)$. Because the
nonorientable 4-genus is bounded by the nonorientable 3-genus, it is known that
$\gamma_4(C(m,n)) \le 3$. By using explicit constructions to obtain upper
bounds on $\gamma_4$ and known obstructions derived from Donaldson's
diagonalization theorem to obtain lower bounds on $\gamma_4$, we produce
infinite subfamilies of $C(m,n)$ where $\gamma_4=0,1,2,$ and $3$, respectively.
However, there remain infinitely many double twist knots where our work only
shows that $\gamma_4$ lies in one of the sets $\{1,2\}, \{2,3\}$, or
$\{1,2,3\}$. We tabulate our results for all $C(m,n)$ with $|m|$ and $|n|$ up
to 50. We also provide an infinite number of examples which answer a conjecture
of Murakami and Yasuhara.
