A model of layered superconductors with exclusively interlayer pairing is studied. It is shown that singlet and triplet interlayer pairing have the same amplitude as long as the hopping between the layers is not too large and the effective interaction is dispersionless. However, a dispersion of the effective interaction results in the splitting of the critical temperature of singlet and triplet pairing for a finite hopping amplitude. It is shown that not only impurities in the layers but also impurities between the layers depress the critical temperature of interlayer pairing. Additionally, it is shown that impurities between the layers scattering electrons hopping from layer to layer lead to a stronger depression of triplet than of singlet interlayer pairing so that the critical temperature is split. We derive expressions for physical properties which are sensitive to the structure of the order parameter: the anisotropic penetration depth, the Josephson current to a conventional superconductor, the magnetic susceptibility which determines the Knight shift, and the jump in the specific heat in the case of the splitting of the transition temperature. We discuss experiments on layered high-temperature superconductors and conclude that the existence of interlayer pairing cannot be excluded in the superconducting state of these compounds.