The independence of partial anhysteretic remanent magnetizations (pARMs) was tested by alternating field demagnetizing orthogonal pARMs whose blocking field intervals do not overlap with each other. Experimentally the two pARMs demagnetized independently for single-domain grains but for multidomain and pseudo-single-domain grains, a superimposed remanence significantly affected the demagnetization of the pre-existing remanence. Experimentally and in numerical simulations, when a total anhysteretic remanent magnetization (simulating primary remanence in a field H a ) was orthogonally overprinted by pARM (simulating remagnetization in H b ), estimation of the intensity of H b was relatively successful but the intensity of H a was always underestimated. Conversely, the direction of H a was preserved but the direction of H b was spurious. These striking results are caused by the violation of the pARM independence law, implying that the conventional sample rejection criteria in pseudo-Thellier paleointensity determination require modification.