  We use the t-J model and Mori projection operator formalism for calculating
the magnetic susceptibility of p-type cuprates in the superconducting and
pseudogap phases. The lack of extended tails in the peaks of the hole spectral
function was shown to provide an incommensurate low-frequency response with
hole dispersions derived from photoemission. The theory reproduces the
hourglass dispersion of the susceptibility maxima with the upper branch
reflecting the dispersion of localized spin excitations and the lower branch
being due to incommensurate maxima of their damping. The intensive resonance
peak appears when the hourglass waist falls below the bottom of the
electron-hole continuum. In the pseudogap phase, the Fermi arcs lead to a
quasi-elastic incommensurate response for low temperatures. This result
explains the lack of the superconducting gap in the susceptibility of
phase-separated underdoped lanthanum cuprates. It may also explain the
strengthening of the quasi-elastic response by magnetic fields and impurities.
The theory accounts for the magnetic stripe reorientation from the axial to
diagonal direction at low hole concentrations.
