We discuss a number of semigroups generated by neutral functional equations of the form \[ d/dt(x(t)+\mu *x(t))+\nu *x(t)=f(t),\quad t\geq 0,\quad x(t)=\phi (t),\quad t\leq 0. \] They are of extended initial function type and of extended forcing function type, and they differ from each other by the amount of smoothness which is imposed on x and f above. The extended initial function type semigroups are adjoints of the extended forcing function type semigroups, and vice versa. The two types of semigroups are also equivalent in the sense that there is a one-to- one, bicontinuous mapping of the state space onto itself, which maps a semigroup of the initial function type onto a semigroup of the forcing function type. In particular, it suffices to study the asymptotic behavior of one of the two types of semigroups, because the results can easily be transferred to the other type of semigroups.