The formalism for tetraatomic reactive scattering using row-orthonormal hyperspherical coordinates is presented. The transformation properties of these coordinates under kinematic rotations and symmetry operations are derived, as are the corresponding Hamiltonian and volume element. Each of the nine operators which contribute to this Hamiltonian is kinematic-rotation invariant. Continuity conditions appropriate for the absence and presence of the geometric phase associated with conical intersections are described. It is concluded that the row-orthonormal hyperspherical coordinates are particularly well suited for calculations of reactive scattering in tetraatomic systems.