Observations of very quiet Sun using the Solar Optical Telescope/Spectro-Polarimeter (SOT/SP) aboard the Hinode spacecraft reveal that the quiet internetwork regions are pervaded by horizontal magnetic flux. The spatial average horizontal apparent flux density derived from wavelength-integrated measures of Zeeman-induced linear polarization is BTapp=55 Mx cm-2, as compared to the corresponding average vertical apparent flux density of |BLapp|=11 Mx cm-2. Distributions of apparent flux density are presented. Magnetic fields are organized on mesogranular scales, with both horizontal and vertical fields showing ``voids'' of reduced flux density of a few granules spatial extent. The vertical fields are concentrated in the intergranular lanes, whereas the stronger horizontal fields are somewhat separated spatially from the vertical fields and occur most commonly at the edges of the bright granules. High-S/N observations from disk center to the limb help to constrain possible causes of the apparent imbalance between |BLapp| and BTapp, with unresolved structures of linear dimension on the surface smaller by at least a factor of 2 relative to the SOT/SP angular resolution being one likely cause of this discrepancy. Other scenarios for explaining this imbalance are discussed. The horizontal fields are likely the source of the ``seething'' fields of the quiet Sun discovered by Harvey et al. The horizontal fields may also contribute to the ``hidden'' turbulent flux suggested by studies involving Hanle effect depolarization of scattered radiation.