We present new spectroscopy and Hubble Space Telescope imaging of protostellar jets discovered in an Hα survey of the Carina Nebula. Near-IR Fe II emission from these jets traces dense gas that is self-shielded from Lyman continuum photons from nearby O-type stars, but is excited by non-ionizing FUV photons that penetrate the ionization front within the jet. New near-IR Fe II images reveal a substantial mass of dense, neutral gas that is not seen in Hα emission from these jets. In some cases, Fe II emission traces the jet inside its natal dust pillar, connecting the larger Hα outflow to the embedded IR source that drives it. New proper motion measurements reveal tangential velocities similar to those typically measured in lower-luminosity sources (100 – 200 km s-1). Combining high jet densities and fast outflow speeds leads to mass-loss rate estimates an order of magnitude higher than those derived from the Hα emission measure alone. Higher jet mass-loss rates require higher accretion rates, implying that these jets are driven by intermediate-mass (∼ 2 – 8 Mspecial character omitted protostars. For some sources, the mid-IR luminosities of the driving sources are clearly consistent with intermediate-mass protostars; others remain deeply embedded and require long-wavelength, high-resolution images to confirm their luminosity. These outflows are all highly collimated, with opening angles of only a few degrees. With this new view of collimated jets from intermediate-mass protostars, we argue that these jets reflect essentially the same outflow phenomenon seen in low-mass protostars, but that the collimated atomic jet core and the material it sweeps up are irradiated and rendered observable. Thus, the jets in Carina offer strong additional evidence that stars up to ∼ 8 Mspecial character omitted form by the same accretion mechanisms as low-mass stars.