  A half-commutative orthogonal Hopf algebra is a Hopf *-algebra generated by
the self-adjoint coefficients of an orthogonal matrix corepresentation
$v=(v_{ij})$ that half commute in the sense that $abc=cba$ for any $a,b,c \in
\{v_{ij}\}$. The first non-trivial such Hopf algebras were discovered by Banica
and Speicher. We propose a general procedure, based on a crossed product
construction, that associates to a self-transpose compact subgroup $G \subset
U_n$ a half-commutative orthogonal Hopf algebra $\mathcal A_*(G)$. It is shown
that any half-commutative orthogonal Hopf algebra arises in this way. The
fusion rules of $\mathcal A_*(G)$ are expressed in term of those of $G$.
