Recently, research about bound states in the continuum (BICs) has become more and more attractive. Nanostructures with rotational symmetry are usually utilized to realize polarization-independent quasi-BIC resonances. Here, we propose a new, to the best of our knowledge, scheme for a polarization-independent quasi-BIC without the help of rotational symmetry. With the rotation of the polarization direction of the incident light, a quasi-BIC resonance can be consistently observed in a dielectric cubic tetramer metasurface without rotational symmetry. Based on far-field multipolar decomposition and near-field electromagnetic distributions, it is found that different multipoles exhibit different dependences on the polarization direction, and the switch between electric and magnetic quadrupoles results in polarization-independent quasi-BIC resonance. Our findings provide an alternative scheme to design polarization-independent devices and promote wider potential applications.