Two competing paradigms of the response of ionospheric convection to changes in the interplanetary magnetic field are synthesized into the common framework of the expanding-contracting polar cap (ECPC) model in order to reconcile apparently contradictory observations reported in the literature over the last 15 years or more. This is done by deriving analytical solutions to the ECPC model for two postulated idealized cases that capture the primary mathematical distinction between the two paradigms, and then simulating how these two responses would appear to different ways of observing them. It is shown that a single type of response could account for most of the variety of observed convection responses. In particular, we show that real phase motions along the polar cap boundary, such as due to an expansion of the reconnection X-line at the magnetopause, would be more readily identified by some observing methods than others, providing an explanation for why authors disagree on whether such phase motions actually exist.