We report the identification of the double degenerate system NLTT 16249 that comprises a normal, hydrogen-rich (DA) white dwarf and a peculiar, carbon-polluted white dwarf (DQ) showing photospheric traces of nitrogen. We disentangled the observed spectra and constrained the properties of both stellar components. In the evolutionary scenario commonly applied to the sequence of DQ white dwarfs, both carbon and nitrogen would be dredged up from the core. The C/N abundance ratio (~ 50) in the atmosphere of this unique DQ white dwarf suggests the presence of unprocessed material (14N) in the core or in the envelope. Helium burning in the DQ progenitor may have terminated early on the red-giant branch after a mass-ejection event leaving unprocessed material in the core although current mass estimates do not favor the presence of a low-mass helium core. Alternatively, some nitrogen in the envelope may have survived an abridged helium-core burning phase prior to climbing the asymptotic giant-branch. Based on available data, we estimate a relatively short orbital period (P