At low temperature, information can be stored in the orientation of the localized magnetic moment of an adatom. However, scattering of electrons and phonons with the nanomagnet leads its state to have incoherent classical dynamics and might cause fast loss of the encoded information. Recently, it has been understood that such scattering obeys certain selection rules due to the symmetries of the system. By analyzing the point-group symmetry of the surface, the time-reversal symmetry and the magnitude of the adatom effective spin, we identify which nanomagnet configurations are to be avoided and which are promising to encode a stable bit. A new tool of investigation is introduced and exploited: the quasispin quantum number. By means of this tool, our results are easily generalized to a broad class of bipartite cluster configurations where adatoms are coupled through Heisenberg-like interactions. Finally, to make contact with the experiments, numerical simulations have been performed to show how such stable configurations respond to typical scanning tunneling microscopy measurements.