We have analysed deep optical spectra of the `born-again' planetary nebula Abell 58 and its hydrogen-deficient knot, surrounding V605 Aql, which underwent a nova-like eruption in 1919. Our analysis shows that the extinction towards the central knot is much higher than previously thought, with c(Hβ) = 2.0. The outer nebula is less reddened, with c(Hβ) = 1.04. We find that the outer nebula has a Ne/O ratio higher than the average PN value.The electron temperature we derive for the central knot varies widely depending on the diagnostic used. The [OIII] nebular-to-auroral transition ratio gives a temperature of 20800K, while the ratio of the [NII] nebular and auroral lines gives Te = 15200K. The helium line ratios λ5876/λ4471 and λ6678/λ4471 imply temperatures of 350 and 550K, respectively. Weakly temperature-sensitive OII recombination line ratios imply similarly low electron temperatures. Abundances derived from recombination lines are vastly higher than those found from collisionally excited lines, with the abundance discrepancy factor (ADF) for O2+ reaching 89 - the second highest known value after that found for the hydrogen-deficient knots in Abell 30. The observed temperature diagnostics and abundances support the idea that, like Abell 30, the knot of Abell 58 contains some very cold ionized material. Although the central star is carbon-rich (C/O > 1), the knot is found to be oxygen-rich, a situation not predicted by the single-star `born-again' theory of its formation. We compare the known properties of Abell 58 to those of Abell 30, Sakurai's Object and several novae and nova remnants. We argue that the abundances in the ejecta observed in A30 and A58 have more in common with neon novae than with Sakurai's Object, which is believed to have undergone a final helium flash. In particular, the C/O ratio of less than unity and the presence of substantial quantities of neon in the ejecta of both Abell 30 and Abell 58 are not predicted by very late thermal pulse models.