The early universe and ultra-relativistic heavy-ion collisions are arenas where QCD matter exists under extreme conditions. Considering the overlap in a generic non-central high-energy nucleus-nucleus collision, the created fireball of quark-gluon plasma (QGP) can sustain strong vorticity due to the finite impact parameter arising from the geometry of the collision and the resultant deposition of angular momentum. This QGP droplet can be embedded in strong magnetic fields that are sourced by the spectator protons of the incident nuclei and likely sustained by the the conductivity and swirling charges in the fluid medium. Such a physical system inspires a suitable extension of the conventional $T$--$\mu_B$ QCD phase diagram. We augment the temperature, $T,$ and baryon chemical potential, $\mu_B,$ axes with one for external magnetic field, $B,$ and another for angular velocity, $\omega$. To achieve this, we identify the confinement-deconfinement transition region of the phase space. This is implemented via two independent routes, one from a rapid rise in scaled entropy density and another dealing with a dip in the squared speed of sound, both within the framework of a modified statistical hadronization model. We find that this approach yields an estimate of the deconfinement temperature $T_C(\mu_B,~\omega,~eB)$ that is found to decrease with increasing $\mu_B,~\omega$ and $eB$. The most prominent drop (by nearly $40$ to $50$ MeV) in $T_C$ occurs when all the three quasi-control (dependent on collision energy, centrality class, etc.) parameters are tuned to values that are achievable in present and upcoming heavy-ion colliders. We also propose some potentially important magneto-rotational consequences on quark-hadron phenomenology such as a reinterpretation of chemical freeze-out data from peripheral collisions in the service of measurements of the fireball's magnetic field (magnetometer) and vorticity (anemometer).