Charged Lepton Flavour Violation is New Physics that must occur, because  neutrinos have masses and mixing angles. In many cases, it can be parametrised at low energy by contact interactions. I focus on the process of mu->e conversion in nuclei, because its experimental sensitivity should improve by 4-6 orders of magnitude in coming years.  The nuclear physics of this process is similar to WIMP scattering: previous calculations of mu-e conversion focused on the (coherent, A^2 enhanced) spin-independent contribution; I will introduce the spin-dependent process, which depends on different operator coefficients,  and  explore  the prospects for distinguishing coefficients (or models) by using different targets.