The Cu3O4 layer in Sr2Cu3O4Cl2 is a variant of the square CuO2 lattice of the high-temperature superconductors, in which the center of every second plaquette contains an extra Cu2+ ion. The ions that make up the conventional CuO2 network, called CuI, have CuI-CuI exchange energy ≈130meV, and order antiferromagnetically at about 380 K; the CuII-CuII exchange is only ≈10meV, and the CuII’s order at ≈40K. A study is reported here of the dependence of the magnetization on field, temperature, and crystallographic orientation for this interesting system. We show that the small permanent ferromagnetic moment, that appears when the CuI spins order, and the unusual spin rotation transitions seen most clearly for one particular direction of the magnetic field, are the result of several small bond-dependent anisotropic terms in the spin Hamiltonian that are revealed because of the frustration of the isotropic Heisenberg interaction between CuI and CuII spins. These include a term which favors collinearity of the CuI and CuII spins, which originates from quantum fluctuations, and also the pseudodipolar interaction. Some of these small interactions also come into play in other lamellar cuprates, connected with the high-Tc superconductivity materials, and in many spin-chain and spin-ladder compounds.