  We study a new family of sign-changing solutions to the stationary nonlinear
Schr\"odinger equation $$ -\Delta v +q v =|v|^{p-2} v, \qquad \text{in
$\mathbb{R}^3$,} $$ with $2<p<\infty$ and $q \ge 0$. These solutions are
spiraling in the sense that they are not axially symmetric but invariant under
screw motion, i.e., they share the symmetry properties of a helicoid. In
addition to existence results, we provide information on the shape of spiraling
solutions, which depends on the parameter value representing the rotational
slope of the underlying screw motion. Our results complement a related analysis
of Del Pino, Musso and Pacard for the Allen-Cahn equation, whereas the nature
of results and the underlying variational structure are completely different.
