Let H be a Hopf algebra over an algebraically closed field of characteristic not 2. Assume that the antipode of H has period 2, and let V be a finite-dimensional representation of H. Then if V is self-dual, it must be either ``orthogonal'' or ``symplectic'' , in the sense that it admits a non-degenerate H-invariant bilinear form which is either symmetric or skew-symmetric. We consider various Hopf algebras constructed from finite groups, and investigate when all of their (fin dim) representations are orthogonal in the above sense.