  We prove the existence of a class of orbitally stable bound state solutions
to nonlinear Schr\"odinger equations with super-quadratic confinement in two
and three spatial dimensions. These solutions are given by time-dependent
rotations of a non-radially symmetric spatial profile which in itself is
obtained via a doubly constrained energy minimization. One of the two
constraints imposed is the total mass, while the other is given by the
expectation value of the angular momentum around the z-axis. Our approach also
allows for a new description of the set of minimizers subject to only a single
mass constraint.
