Let (G; ) and (G0;
) be groupoids. A bijection f : G ! G0 is called a halfisomorphism if f(x  y) 2 ff(x)
f(y); f(y)
f(x)g, for any x; y 2 G. A half-isomorphism of a groupoid onto itself is a half-automorphism. A half-isomorphism f is called special if f^-1 is also a half-isomorphism. In this paper, necessary and sufficient conditions for the existence of special half-isomorphisms on groupoids and quasigroups are obtained. Furthermore, some examples of non-special half-automorphisms for loops of infinite order are provided.