Self-interacting Dark Matter (SIDM) has been proposed as a way to help reconcile small scale astrophysical observations with CDM predictions. We use N-body simulations to study the effect of SIDM on the morphology of disk galaxies falling into galaxy clusters. An effective drag force arises from dark matter scatterings and leads to offsets of the stellar disk with respect to its surrounding halo, causing distortions in the disk. We show that potentially observable warps, asymmetries, and thickening of the disk occur in simulations with currently allowed cross-sections. With further analysis of the potential systematic uncertainties of these novel probes, we believe it could be possible to constrain SIDM cross-sections with current and future observations.