A characteristic pattern of solar hard X-ray emission, first identified in SOL1969-03-30 by Frost & Dennis (1971), turns out to have a close association with the prolonged high-energy gamma-ray emission originally observed by Forrest et al. (1985). This identification has become clear via the observations of long-duration γ-ray flares by the Fermi/LAT experiment, for example in the event SOL2014-09-01. The distinctive features of these events include flat hard X-ray spectra extending well above 100 keV, a characteristic pattern of time development, low-frequency gyrosynchrotron peaks, CME association, and gamma-rays identifiable with pion decay originating in GeV ions. The identification of these events with otherwise known solar structures nevertheless remains elusive, in spite of the wealth of EUV imagery available from SDO/AIA. The quandary is that these events have a clear association with SEPs in the high corona, and yet the gamma-ray production implicates the photosphere itself, despite the strong mirror force that should focus the particles away from the Sun We discuss the morphology of these phenomena and propose a solution to this problem.