We report the results of an extensive investigation of the magnetic properties of a large series of undoped R2CuO4 single crystals with R==Pr, Nd, Sm, Eu, and Gd (which are the host compounds for the newly discovered series of electron cuprate superconductors) and mixture versions of the form A2-xBxCuO4, with A==Pr, Nd, Sm, Eu, or Gd, and B==Gd, Tb, or Dy. We have measured dc and ac magnetization, microwave magnetoabsorption, EPR, and specific heat. These measurements reveal two characteristic transition temperatures associated with a novel complex magnetic behavior, including weak ferromagnetism, two sharp peaks in the low-field dc magnetization, an unusual anisotropy in the EPR resonance field for R=Gd, and two additional anisotropic microwave absorption modes. The higher characteristic transition temeperature at 270 K is associated with antiferromagnetic ordering of the Cu moments which are strongly coupled within the CuO2 layers. The lower, at 20 K, cannot be attributed to antiferromagnetic ordering of the R moments and is tentatively attributed to a spontaneous canted spin reorientation. An understanding of this magnetic behavior is important in order to ascertain its relationship to possible mechanisms of high-temperature superconductivity. © 1990 The American Physical Society.