We obtained optical/near-IR rest-frame Magellan FIRE spectra (including Pa$\beta$ and Pa$\gamma$) of 25 starburst galaxies at 0.5<z<0.9, with average star formation rates (SFR) x7 above the Main Sequence (MS). We find that Paschen-to-Balmer line ratios saturate around a constant value corresponding to $A_{\rm V}\sim$2-3 mag, while line to IR luminosity ratios suggest a large range of more extreme obscurations and appear to be uncorrelated to the former. This behavior is not consistent with standard attenuation laws derived for local and distant galaxies, while being remarkably consistent with observations of starburst cores in which young stars and dust are homogeneously mixed. This model implies $A_{\rm V}=$2-30magattenuationtothecenterofstarburstcores,withamedianof 9mag(afactorof4000).X-rayhardnessratiosfor6AGNsinoursampleandcolumndensitiesderivedfromobserveddustmassesandradiosizesindependentlyconfirmthislevelofattenuation.Intheseconditionsobservedoptical/near-IRemissioncomesfromsurfaceregions,whileinnerstarburstcoresareinvisible.Wethusattributethehigh[NII]/H$\alpha$ ratios to widespread shocks from accretion, turbulence and dynamic disturbances rather than to AGNs. The large range of optical depths demonstrates that substantial diversity is present within the starburst population, possibly connected to different merger phases or progenitor properties. The majority of our targets are, in fact, morphologically classified as mergers. We argue that the extreme obscuration provides in itself smoking gun evidence of their merger origin, and a powerful tool for identifying mergers at even higher redshifts.