Using the time-dependent Lagrangian response approach, the recently revived orbital optimized coupled cluster (OCC) model is reformulated using nonorthogonal orbital rotations in a manner that conserves the commutativity of the cluster excitation operators. The gauge invariance and the simple pole structure of the OCC linear response function are retained, while the dimension of the eigenvalue problem is reduced by a factor of 2. Restricting the cluster operator to double excitations, we have carried out the first implementation of gauge invariant coupled cluster response theory. Test calculations of the excitation energy, and length and velocity gauge oscillator strengths are presented for the lowest electric dipole allowed transitions of the CH+ molecular ion and the Ne atom. Additionally, the excitation energies to the four lowest-lying states of water are calculated.