Low-temperature cycling (LTC) of magnetic remanences carried by rocks has become a standard technique in paleomagnetism, rock magnetism, and environmental magnetism as a means of identifying mineralogy and grain size. LTC usually involves measuring low-temperature thermomagnetic curves on cooling through crystallographic transitions, such as magnetite's Verwey transition. Historically, it has been assumed that remanence carried by single-domain (SD) magnetite grains is not affected by cooling through the cubic/monoclinic Verwey transition, whereas larger multidomain (MD) magnetite grains partially demagnetize. However, it has been recently pointed out that the shape anisotropy even for an infinitely long cylinder is approximately 3 times smaller than the monoclinic magnetocrystalline anisotropy along the hardest axis, i.e., SD remanences are not impervious to LTC. Using a micromagnetic algorithm we simulate LTC curves for assemblages of effectively elongated SD magnetite grains and consider the contribution of magnetostatic interactions. Initially, we assume that relationship between the cubic and monoclinic symmetry is chosen randomly; however, there are key experimental features, which this model does not explain. A new "controlled switching" model is developed; the orientation of the low-temperature monoclinic axes are not chosen randomly, but instead are controlled by the direction of the magnetic moment on cooling through the Verwey transition. This new model correctly predicts experimentally observed low-temperature trends that the "random" model does not. We therefore propose a new model for the mechanism controlling the behavior of SD grains at the Verwey transition and show that the low-temperature behavior of SD and MD grains can yield ambiguously similar behavior.