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In a system of C2 symmetry, symmetry-protected bound states in the continuum (SP-BICs) exist with a continuous spectrum of radiating waves that can carry energy away and enable an infinite radiative quality (Q) factor and zero linewidth. However, the SP-BICs transform into quasi-BICs by breaking the C2 symmetry of the system, where the resonance lifetime and linewidth become finite and measurable. As such, the quasi-BICs are very sensitive to the polarization of incident radiation. Owing to the application of a biosensor or a lasing device, it is unavoidable to work with an unpolarized radiative beam. Herein, we propose a metasurface in a C4 symmetric layout, which exhibits polarization-insensitive terahertz symmetry-protected quasi-BICs. The orientations of adjacent two meta-molecules (MMs) are designed to be orthogonal to each other. By tuning the degree of asymmetry along the orientation of MMs, the quasi-BICs exhibit insensitivity to the polarization of the incident terahertz wave. A large degree of asymmetry results in a deformation of an electric quadrupole, which forms an energy leaky channel to the free space. Due to the translational symmetry, the wave-vector of the lattice in C4 symmetry is conserved so that the electric components of transmitted radiation along the x axis is identical to that along the y axis, Txx = Tyy. As such, the leaky channel of electromagnetic scattering becomes insensitive to the incident polarization. Our results present an approach to achieve polarization-insensitive quasi-BICs in a topologically symmetric metasurface, which is helpful for the innovation of terahertz biosensor. [ABSTRACT FROM AUTHOR]