We propose a theoretical quantum model and derive a set of analytic formulas to study the physical properties of a pair of double-walled magnetic nanotubes. The Heisenberg exchange parameters between the two walls of the nanotubes are assumed to differ only in sign. Thus, in the absence of external magnetic field, our calculated macroscopic properties of this pair of nanotubes are almost precisely identical, exhibiting fascinating duality of the nanosystems and demonstrating the correctness of our theoretical model. The two spin systems are all frustrated, so that sudden changes in the macroscopic properties are observed around T that is well below the transition temperature T. However, only the inner shell consisting of smaller A-type spins has been obviously affected. In the temperature range T < T < T, this shell becomes semi-antiferromagnetic and its magnetization is considerably suppressed, whereas as temperature falls below T the shell gradually restores its ferromagnetic nature. The longitudinal hysteresis behavior of such a double-waled nanotube is ferromagnetic-like below T, but antiferromagnetic-like in the temperature interval T < T < T. Moreover, we find that the diameter of the nanotube has seemly no effects on its physical properties, whereas its length does affects the two temperatures slightly, and also its spin configuration at very low temperatures if the tube is sufficiently long. More importantly, the theoretical results presented in this paper can be precisely reproduced with the quantum computational method we develop in recent years, justifying the validity of the numerical approach once again.