The lanthanum family of high-temperature cuprate superconductors is known to exhibit both spin and charge electronic modulations around a doping level of 1/8. We assume that these modulations have the character of a two-dimensional spin-vortex checkerboard and investigate whether this assumption is consistent with the Fermi surface and the pseudogap measured by angle-resolved photoemission spectroscopy. We also explore the possibility of observing quantum oscillations of transport coefficients in such a background. These investigations are based on a model of noninteracting spin-1/2 fermions hopping on a square lattice and coupled through spins to a magnetic field imitating a spin-vortex checkerboard. The main results of this paper include (i) a calculation of the Fermi surface containing Fermi arcs at the positions in the Brillouin zone largely consistent with experiments, (ii) identification of factors complicating the observations of quantum oscillations in the presence of spin modulations, and (iii) an investigation of the symmetries of the resulting electronic energy bands, which, in particular, indicates that each band is doubly degenerate and has at least one conical point, where it touches another doubly degenerate band. We discuss possible implications these cones may have for the transport properties and the pseudogap.