We study the physical conditions, elemental abundances, and kinematics of the high-velocity clouds (HVCs) along the sight lines toward active galaxies HE0226-4110 and PG0953+414 using Hubble Space Telescope Imaging Spectrograph and Far Ultraviolet Spectroscopic Explorer data. Our observations reveal multiple components of HVC absorption in lines of HI, CII, CIII, CIV, OVI, SiII, SiIII, and SiIV in both directions. We investigate whether photoionization by the extragalactic background radiation or by escaping Milky Way radiation can explain the observed ionization pattern. We find that photoionization is a good explanation for the CII, CIII, SiII, and SiIII features, but not for the OVI or CIV associated with the HVCs, suggesting that two principal phases exist: a warm (T~10^4K), photoionized phase and a hotter (T=1-3x10^5K), collisionally-ionized phase. The warm HVCs toward HE0226-4110 have high levels of ionization (97-99%), and metallicities ([Z/H] between -0.9 and -0.4) close to those in the Magellanic Stream, which lies eleven degrees away on the sky at similar velocities. These HVCs have thermal pressures that would place them close to equilibrium in a fully ionized 10^6 K Galactic corona with n_H=4-9x10^{-5}cm^{-3} at 50 kpc. A mini-survey of the hot, collisionally ionized HVC components seen here and in five other sight lines finds that in 11/12 cases, the high ions have kinematics and ionic ratios that are consistent with an origin in conductive interfaces. However, the broad absorption wing on the OVI profile toward PG0953+414 is not completely explained by the interface scenario, and may be tracing the outflow of hot gas into the Milky Way halo as part of a Galactic fountain or wind.
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