At times, relativistic electron fluxes in Earth's outer magnetosphere are not obviously related to an external (Jovian or solar) source. This finding suggests that an internal magnetospheric acceleration mechanism may operate under some circumstances. A possible mechanism identified for Jupiter's magnetosphere could also be considered in the terrestrial case. Such a model requires the substorm- generation of a spectrally-soft electron component with subsequent inward radial diffusion (violating the third adiabatic invariant). A large electron energy gain transverse to the magnetic field occurs in this process. Eventually, deep within the magnetosphere, substantial pitch angle scattering occurs violating all adiabatic invariants. Then, at low L-values, there occurs an energy-preserving outward transport of energetic electrons near the mirror points. This leads to a return of the accelerated population to the outer magnetosphere. Such low-altitude processes should result in ''conic'' or ''butterfly'' pitch angle distributions at very high energies as the electrons execute trans-L diffusion at the mirror altitudes and then are magnetically focussed near the equator. Data collected concurrently at geostationary orbit at three widely-spaced local times during a relativisic electron event show a butterfly pitch angle distribution, while lower energy electrons simultaneously show pancake-like distributions. The butterfly pitch angle distributions appear in ~25% of the examined relativistic electron events, thereby providing support for acceleration by a recirculation process. ¿ American Geophysical Union 1989