We report the refinement of x-ray powder diffraction together with magnetic and thermal conductivity measurements made on the entire family of RMnO3 perovskites prepared by melt growth or under high pressure. Analysis of the data has identified the origin of the transition from type-A to type-E magnetic order as a competition between t-orbital and e-orbital spin-spin interactions within each Mn-O-Mn bond in the (001) planes, the e-orbital interactions decreasing with decreasing R3+-ion size.