Periodic leaky-wave antennas (PLWAs) suffer two main issues limiting their scanning ability. The first one is n= −2 spatial harmonic radiation, generating high grating-lobe levels. In this paper, a simple technique by introducing glide-symmetry structure to the PLWA is analysed and used to reduce n = −2 space harmonic level and enhance n = −1 harmonic radiation, subsequently. The other issue is open-stopband (OSB), preventing PLWAs from scanning through broadside. By analysing the ABCD matrix of the cascaded equivalent circuit for the unit cell, a conjugate match condition to counteract the susceptance of the radiation element to suppress the OSB is obtained, which is simple without extracting any parameters. A half-mode substrate integrated waveguide-based prototype is designed and manufactured to demonstrate the proposed methods. The measured results show that the proposed antenna achieves gain fluctuation lower than 1 dB and sidelobe level less than −12 dB, indicating OSB and n = −2 harmonic suppression well, which demonstrates the validity of the proposed method.