  We investigate the quantum phases of the frustrated spin-$\frac{1}{2}$
$J_1$-$J_2$-$J_3$ Heisenberg model on the square lattice with ferromagnetic
$J_1$ and antiferromagnetic $J_2$ and $J_3$ interactions. Using the
pseudo-fermion functional renormalization group technique, we find an
intermediate paramagnetic phase located between classically ordered
ferromagnetic, stripy antiferromagnetic, and incommensurate spiral phases. We
observe that quantum fluctuations lead to significant shifts of the spiral
pitch angles compared to the classical limit. By computing the response of the
system with respect to various spin rotation and lattice symmetry-breaking
perturbations, we identify a complex interplay between different nematic spin
states in the paramagnetic phase. While retaining time-reversal invariance,
these phases either break spin-rotation symmetry, lattice-rotation symmetry, or
a combination of both. We therefore propose the $J_1$-$J_2$-$J_3$ Heisenberg
model on the square lattice as a paradigmatic example where different
intimately connected types of nematic orders emerge in the same model.
