We obtained optical long-slit spectra of four planetary nebulae (PNe) with low-ionization pair of knots, namely He 1-1, IC 2149, KjPn 8 and NGC 7662. These data allow us to derive the physical parameters and excitation of the pairs of knots, and those of higher ionization inner components of the nebulae, separately. Our results are as follows. (1) The electron temperatures of the knots are within the range 9500-14500 K, similar to the temperatures of the higher ionization rims/shells. (2) Typical knots' densities are 500-2000 cm-3. (3) Empirical densities of the inner rims/shells are higher than those of the pairs of knots, by up to a factor of 10. Theoretical predictions, at variance with the empirical results, suggest that knots should be denser than the inner regions, by at least a factor of 10. (4) Empirical and theoretical density contrasts can be reconciled if we assume that at least 90 per cent of the knots' gas is neutral (likely composed of dust and molecules). (5) By using the new Raga et al. shock modelling and diagnostic diagrams appropriated for spatially resolved PNe, we suggest that high-velocity shocked knots travelling in the photoionized outer regions of PNe can explain the emission of the pairs of knots analysed in this paper.