We provide a new collection of spectra of 35 carbon stars obtained with the ESO/VLT X-Shooter instrument as part of the X-Shooter Spectral Library project. The spectra extend from 0.3 μm to 2.4 μm with a resolving power above ~8000. The sample contains stars with a broad range of (J - K) color and pulsation properties located in the Milky Way and the Magellanic Clouds. We show that the distribution of spectral properties of carbon stars at a given (J - K) color becomes bimodal (in our sample) when (J - K) is larger than about 1.5. We describe the two families of spectra that emerge, characterized by the presence or absence of the absorption feature at 1.53 μm, generally associated with HCN and C2H2. This feature appears essentially only in large-amplitude variables, though not in all observations. Associated spectral signatures that we interpret as the result of veiling by circumstellar matter, indicate that the 1.53 μm feature might point to episodes of dust production in carbon-rich Miras. Based on observations collected at the European Southern Observatory, Paranal, Chile, Prog. ID 084.B-0869(A/B), 085.B-0751(A/B), 189.B-0925(A/B/C/D).Tables 1, B.1, E.1, E.2 are also available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/589/A36The reduced spectra are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/589/A36