We consider the NLS equation with a linear double-well potential. Symmetry breaking, i.e. the localisation of an order parameter in one of the potential wells that can occur when the system is symmetric, has been studied extensively. However, when the wells are asymmetric, only a few analytical works have been reported. Using double Dirac delta potentials, we study rigorously the effect of such asymmetry on the bifurcation type. We show that the standard pitchfork bifurcation becomes broken, i.e. unfolded, and instead a saddle-centre type is obtained. Using a geometrical approach, we also establish the instability of the corresponding solutions along each branch in the bifurcation diagram. © 2020, Springer Nature B.V.