The exciton dynamics in transverse magnetic field is investigated both experimentally and theoretically in two-monolayer-thick GaAs/AlAs quantum wells with an indirect band gap and a type-II band alignment. The observed linear polarization of the quantum well photoluminescence has two contributions, one of which arises from the crystalline structure of the quantum well. It does not depend on temperature and demonstrates a strong spectral dependence across the emission band. The other one is induced by a transverse magnetic field. It strongly decreases with increasing temperature, has no spectral dependence, and demonstrates an unexpectedly long-time dynamics. The experimental findings can be explained in the framework of the developed theoretical model which accounts for the quantum well anisotropy, the Zeeman effect of electrons and holes in the transverse magnetic field, and the redistribution of excitons over the spin sublevels. It provides quantitative agreement with the experiment and allows us to evaluate, for the studied structure, the heavy-hole in-plane g-factor tensor, which turns out to be extremely anisotropic with principal values of opposite signs and the same magnitude of 0.25.