Type-IV superconductivity: Can superconductivity be more exotic than unconventional?

Abrikosov vortex phase in s-wave layered superconductor in a parallel magnetic field is shown to be absolutely unstable with respect to the appearance of a triplet component which breaks inversion (parity) and spin-rotational symmetries of Cooper pairs. Symmetry breaking paramagnetic effects are demonstrated to be of the order of unity if the orbital upper critical field, H _c2 (0), is of the order of Clogston paramagnetic limiting field, H _p. We suggest a generic phase diagram of such type-IV superconductor, which is singlet one at H=0 and in the Meissner phase and characterized by mixed singlet-triplet order parameter with broken symmetries of Copper pairs in vortex phase. A possibility to observe type-IV superconductivity in clean organic, high-Tc, and MgB ₂ superconductors is discussed.