We consider quantum tunnelling in asymmetric double-well systems for which the local minima in the two wells have the same energy, but the frequencies differ slightly. In a molecular context, this situation can arise if the symmetry is broken by isotopic substitutions. We derive a generalization of instanton theory for these asymmetric systems, leading to an semiclassical expression for the tunnelling matrix element and hence the energy-level splitting. We benchmark the method using a set of one-and two-dimensional models, for which the results compare favourably with numerically exact quantum calculations. Using the ringpolymer instanton approach, we apply the method to compute the level splittings in various isotopomers of malonaldehyde in full dimensionality and analyse the relative contributions from the zero-point energy difference and tunnelling effects.