  We focus on a broad class of tetragonal itinerant systems sharing a tendency
towards the spontaneous formation of incommensurate magnetism with ordering
wavevectors $\mathbf{Q}_{1,2}=(\pi-\delta,0)/(0,\pi-\delta)$ or
$\mathbf{Q}_{1,2}=(\pi,\delta)/(-\delta,\pi)$. Employing a Landau approach, we
obtain the generic magnetic phase diagram and identify the leading
instabilities near the paramagnetic-magnetic transition. Nine distinct magnetic
phases exist that either preserve or violate the assumed $C_4$-symmetry of the
paramagnetic phase. These are single- and double-$\mathbf{Q}$ phases consisting
of magnetic stripes, helices and whirls, either in an individual or coexisting
manner. These nine phases can be experimentally distinguished by polarized
neutron scattering, or, for example, by combining measurements of the induced
charge order and magnetoelectric coupling. Within two representative
five-orbital models, suitable for BaFe$_2$As$_2$ and LaFeAsO, we find that the
incommensurate magnetic phases discussed here are accessible in iron-based
superconductors. Our investigation unveils a set of potential candidates for
the unidentified $C_2$-symmetric magnetic phase that was recently observed in
Ba$_{1-x}$Na$_x$Fe$_{2}$As$_{2}$. Among the phases stabilized we find a
spin-whirl crystal, which is a textured magnetic $C_4$-symmetric phase. The
possible experimental observation of textured magnetic orders in iron-based
superconductor, opens new directions for realizing intrinsic topological
superconductors.
