We introduce a microscopic model aimed at describing the behavior of fermionic excitations in the background of a magnetic texture called a "spin-vortex checkerboard." This texture was proposed previously as a possible alternative to stripes to interpret the experimental phenomenology of spin and charge modulations in 1/8-doped lanthanum cuprates. The model involves two kinds of interacting fermionic excitations residing in spin-rich and spin-poor regions of the modulated structure. It is a generalization of another model developed earlier for the so-called "grid checkerboard." The principal terms of our model describe the decay of fermionic pairs belonging to spin-poor regions into single fermions occupying spin-rich regions and vice versa. These terms induce intricate fermionic correlations throughout the system but fall short of inducing superconductivity unless arbitrarily small hopping terms are added to the model Hamiltonian. We present the mean-field solution of the model, including, in particular, the temperature dependence of the energy gap. The latter is found to be in good overall agreement with available experimental data for high-Tc cuprate superconductors.