In this thesis, we probe the fate of inversion symmetry in the pseudogap phase of cuprate high-temperature superconductors. We study the photogalvanic effect in both single and double layer bismuth-based cuprates with an emphasis on both the circular and linear photogalvanic effects. Through a detailed analysis of photogalvanic currents on the (i) polarization of excitation light, (ii) temperature, (iii) angle of incidence, and (iv) orientation of crystals, we find that the pseudogap state breaks inversion symmetry and exhibits a chiral behavior at the length scale of micrometers. We also carry out a full symmetry analysis on photogalvanic tensors, and find that the point group symmetry of the pseudogap is either mm2 (antiferromagnetic) or mm21 (paramagnetic). With domain averaging that maintains chirality, these symmetry groups are consistent with previous measurements that specifically probe rotation symmetry breaking, time-reversal symmetry breaking, or inversion symmetry breaking, and also with some models of the pseudogap order parameter. The origin of such domain averaging is yet to be understood.