The appearance of spin-density-wave (SDW) magnetic order in the low-temperature and high-field corner of the superconducting phase diagram of CeCoIn₅ is unique among unconventional superconductors. The nature of this magnetic Q phase is a matter of current debate. Here, we present the thermal conductivity of CeCoIn₅ in a rotating magnetic field, which reveals the presence of an additional order inside the Q phase that is intimately intertwined with the superconducting d-wave and SDW orders. A discontinuous change of the thermal conductivity within the Q phase, when the magnetic field is rotated about antinodes of the superconducting d-wave order parameter, demands that the additional order must change abruptly, together with the recently observed switching of the SDW. A combination of interactions, where spin-orbit coupling orients the SDW, which then selects the secondary p-wave pair-density-wave component (with an average amplitude of 20% of the primary d-wave order parameter), accounts for the observed behavior.