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We present stellar specific angular momentum j∗ measurements of two z ∼ 1.5 galaxies in the KGES sample and 12 DYNAMO z ∼ 0.1 analogues of high-redshift galaxies. We combine natural seeing integral field spectroscopic data to trace line emission out to high multiples of effective radius re, with adaptive optics assisted Keck/OSIRIS observations to trace the rapid rise in rotation curve in the inner regions. Our spaxel-wise integration method gives results that are on average within measurement uncertainty of the traditional rotation curve model method. At z ∼ 0, combining GMOS and OSIRIS data sets improves the measurement uncertainty in j∗ from 13 per cent (GMOS only) or 16 per cent (OSIRIS only) to 10 per cent. At z ∼ 1.5, systematics allow for at best 20 per cent uncertainty on j∗. DYNAMO analogues of high-z galaxies have low j∗ for their stellar mass M∗, and low bulge-to-total light ratio β for their j∗/M∗. The high-z galaxy COSMOS 127977 has j∗/M∗ consistent with normal local disc galaxies, while UDS 78317 is consistent with local analogues. However, our high-resolution OSIRIS data reveal that UDS 78317 may be a merging system. We report a relationship between distance to the β−j∗/M∗ plane and the ratio of velocity dispersion to rotational velocity σ/vmax, where galaxies that deviate more from the plane are more dispersion-dominated due to turbulence. Much of the scatter in M∗−j∗ that is not explained by variations in the bulge-to-total ratio or evolution with redshift may be driven by increased turbulence due to star formation, or by treating mergers as rotating discs.