Reversed structures of artificial spin-ice systems, where elongated holes with elliptical shape (antidots) are arranged into a square arraywith two orthogonal sublattices, are referred to as antisquared spin ice. UsingBrillouin light-scattering spectroscopy and plane-wave-method calculations, we investigate the spin-wave propagation perpendicular to the applied field direction for two 20-nm-thick Permalloy nanostructures which differ by the presence of single- and double-elliptical antidots. For the spin-wave propagation along the principal antidot lattice axis, the spectrum consists of flat bands separated by several frequency gaps which are the effect of spin-wave amplitude confinement in the regions between antidots. Contrarily, for propagation direction at 45° with respect to the antidot symmetry axis, straight and narrow channels of propagation are formed, leading to broadening of bands and closing of the magnonics gaps. Interestingly, in this case, extra magnonic band gaps occur due to the additional periodicity along this direction. The width and the position of these gaps depend on the presence of single or double antidots. In this context, we discuss possibilities for the tuning of spin-wave spectra in antisquared spin-ice structures.