A sub-population of energetic rotation-powered pulsars show high fluxes of pulsed non-thermal hard X-ray emission. While this `MeV pulsar' population includes some radio-loud pulsars like the Crab, a significant number have no detected radio or GeV emission, a mystery since gamma-ray emission is a common characteristic of pulsars with high spin-down power. Their steeply rising hard X-ray spectral energy distributions (SEDs) suggest peaks at 0.1 - 1 MeV but they have not been detected above 200 keV. Several upcoming and planned telescopes may shed light on the MeV pulsars. The Neutron star Interior Composition ExploreR (NICER) will observe pulsars in the 0.2 - 12 keV band and may discover additional MeV pulsars. Planned telescopes, such as All-Sky Medium-Energy Gamma-Ray Observatory (AMEGO) and e-ASTROGAM, will detect emission above 0.2 MeV and polarization in the 0.2 - 10 MeV band. We present a model for the spectrum and polarization of MeV pulsars where the X-ray emission comes from electron-positron pairs radiating in the outer magnetosphere and current sheet. This model predicts that the peak of the SED increases with surface magnetic field strength if the pairs are produced in polar cap cascades. For small inclination angles, a range of viewing angles can miss both the radio pulse and the GeV pulse from particles accelerating near the current sheet. Characterizing the emission and geometry of MeV pulsars can thus provide clues to the source of pairs and acceleration in the magnetosphere.