  We experimentally demonstrate an alternative method for the dynamic
generation of atomic spin squeezing, building on the interplay between linear
coupling and nonlinear phase evolution. Since the resulting quantum dynamics
can be seen as rotation and shear on the generalized Bloch sphere, we call this
scheme twist-and-turn (TnT). This is closely connected to an underlying
instability in the classical limit of this system. The short-time evolution of
the quantum state is governed by a fast initial spreading of the quantum
uncertainty in one direction, accompanied by squeezing in the orthogonal axis.
We find an optimal value of $\xi_{\text{S}}^2=-7.1(3)$ dB in a single BEC and
scalability of the squeezing to more than 10^4 particles with
$\xi_{\text{S}}^2=-2.8(4)$ dB.
