While the standard X-ray variability of black hole X-ray binaries (BHXBs) is stochastic and noisy, there are two known BHXBs that exhibit exotic “heartbeat”-like variability in their lightcurves: GRS 1915+105 and IGR J17091–3624. In 2022, IGR J17091–3624 went into outburst for the first time in the NICER/NuSTAR era. These exquisite data allow us to simultaneously track the exotic variability and the corresponding spectral features with unprecedented detail. We find that as in typical BHXBs, the outburst began in the hard state, then continued in the intermediate state, but then transitioned to an exotic soft state, where we identify two types of heartbeat-like variability (Class V and a new Class X). The flux energy spectra show a broad iron emission line due to relativistic reflection when there is no exotic variability, and absorption features from highly ionized iron when the source exhibits exotic variability. Whether absorption lines from highly ionized iron are detected in IGR J17091–3624 is not determined by the spectral state alone, but rather is determined by the presence of exotic variability; in a soft spectral state, absorption lines are only detected along with exotic variability. Our finding indicates that IGR J17091–3624 can be seen as a bridge between the most peculiar BHXB GRS 1915+105 and “normal” BHXBs, because it alternates between the conventional and exotic behaviors of BHXBs. We discuss the physical nature of the absorbing material and exotic variability in light of this new legacy data set.