The complexity of the phase diagram of the cuprates goes well beyond its unique high-Tc superconducting state, as it hosts a variety of different electronic phenomena, such as the pseudogap, nematic order, charge order, and strange metallic behavior. The parent compound, however, is well understood as a Mott insulator, displaying quenched charge degrees of freedom and low-energy antiferromagnetic excitations described by the Heisenberg exchange coupling J. Here we show that doping holes in the oxygen orbitals inevitably generates another spin interaction - a biquadratic coupling - that must be included in the celebrated t−J model. While this additional interaction does not modify the linear spin wave spectrum, it promotes an enhanced nematic susceptibility that is peaked at a temperature scale determined by J. Our results explain several puzzling features of underdoped YBa2Cu3O7, such as the proximity of nematic and antiferromagnetic order, the anisotropic magnetic incommensurability, and the in-plane resistivity anisotropy. Furthermore, it naturally accounts for the absence of nematicity in electron-doped cuprates, and supports the idea that the pseudogap temperature is related to strong local antiferromagnetism.