We argue that it may be possible to exploit neutrinos from the CN cycle and p-p chain to determine the primordial solar core abundances of C and N at an interesting level of precision. Such a measurement would allow a comparison of the Sun's deep interior composition with its surface, testing a key assumption of the standard solar model (SSM), a homogeneous zero-age Sun. It would also provide a cross-check on recent photospheric abundance determinations that have altered the once excellent agreement between the SSM and helioseismology. As further motivation, we discuss a speculative possibility in which the photospheric abundance-helioseismology puzzle is connected with the solar system metal differentiation that accompanied formation of the gaseous giant planets. The theoretical relationship between core C and N and the 13N and 15O solar neutrino fluxes can be made more precise (and more general) by making use of the Super-Kamiokande and Sudbury Neutrino Observatory (SNO)8B neutrino capture rates, which calibrate the temperature of the solar core. The primordial C and N abundances can then be obtained from these neutrino fluxes and from a product of nuclear rates, with little residual solar model dependence. We describe some of the recent experimental advances that could allow this comparison to be made (theoretically) at the ~9% level, and we note that this uncertainty may be reduced further as a result of ongoing work on the S-factor for 14N(p, γ). The envisioned measurement might be possible in deep, large-volume detectors using organic scintillator, for example, Borexino or SNO+.