One of the greatest theoretical challenges in the build-up to the advanced gravitational-wave detector era is the modelling of generic binary waveforms. We introduce an approximation that has the potential to significantly simplify this problem. We show that generic precessing-binary inspiral waveforms (covering a 7-dimensional parameter space) can be mapped to only a two-dimensional space of non-precessing binaries, characterized by mass ratio and a single effective total spin. The mapping consists of a time-dependent rotation, and is extremely accurate (matches > 0.99 with parameter biases in the total spin of less than 0.02). Finally, we outline a general proposal for the construction of precessing-binary waveform models, which will be the focus of future work.