  In this paper, using different approaches we demonstrate the equality of the
two mixed charge-spin response functions of a spin-polarized electron gas when
orbital effects are negligible. Within a generalized STLS approximation we show
that the two mixed responses are equal. We also present arguments for the
equality of the two dynamic responses by considering a symmetry of the
effective screened interaction between two opposite spin electrons.
Furthermore, using the reflection symmetry of the system and the fact that the
hamiltonian is real we prove rigorously that the two responses coincide
identically.
