Argumentation theory emphasizes that agreement at some level is a prerequisite for meaningful disagreement. But what about disagreements that are so profound and go so deeply that the advocates find no basis for underlying agreement? In those cases there may be no dialectical or logical means to resolve the impasse. But rhetorical resolutions may be available if audiences can be convinced to perceive the argument in a new way. This essay identifies four pairs of rhetorical moves—inconsistency, packaging, time, and shifting the ground—that might be employed, and then develops two extended examples: one involving Lyndon Johnson’s arguments for federal aid to education, which concluded successfully; and one on my own arguments about abortion, which ended in failure. This essay originally was presented at the Seventh Conference of the International Society for the Study of Argumentation, held in Amsterdam in 2010. It is reprinted from Topical Themes in Argumentation Theory (Frans H. van Eemeren and Bart Garssen, Ed.), pp. 77–89, published by Springer in 2012.