A flip on an invertible dynamical system \(T\colon X\to X\) is an involution \(F\colon X\to X\) satisfying \(F\circ T=T^{-1}\circ F\). A flip system \((X,T,F)\) can be thought as an action of the infinite dihedral group \(D_{\infty}\) with the infinite cyclic part corresponding to the action of \(T\) and the involution corresponding to \(F\). Here the special case where \(T\) is a shift of finite type and \(F\) is a homeomorphism on the shift space is considered, and the asymptotic behaviour of the orbit counting function is studied. The arguments are combinatorial, and more refined asymptotics are obtained than the ones known for orbit-counting problems for other group actions.