We present new CCD narrowband images and spatially resolved intermediate-dispersion spectroscopy of the hydrogen-poor nebulae A30 and A58. These data have allowed us to obtain information on the physical properties and chemical abundances of the inner knots and outer envelopes of A30 and A58 as a basis for the discussion of their implications in the evolutionary theories. In particular, hydrogen emission from the central knots has been carefully measured, and, consequently, the rates of conversion of hydrogen into helium have been established with accuracy. The rates found indicate that almost all the original hydrogen (between 75% and 95%) has been burned into helium. The chemical segregation among the knots of A30 previously suggested by Jacoby & Ford is confirmed. The A58 inner knot, which is spatially resolved in two different components, shows evidence of excitation by shocks. Helium abundances and the N/O ratio in the outer envelope of this nebula are very high, therefore indicating that the nebula formed as the result of the evolution of an intermediate-mass star. An analysis of the chemical abundances, including other objects of the same class, gives some hints that He/H and 0/H are anticorrelated. The time elapsed between the formation of the outer envelopes and the ejection of the central knots also seems to be related to the central knots' chemical abundances.