  Let $(G,*)$ and $(G',\cdot)$ be groupoids. A bijection $f: G \rightarrow G'$
is called a half-isomorphism if $f(x*y)\in\{f(x)\cdot f(y),f(y)\cdot f(x)\}$,
for any $ x, y \in 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.
