ABSTRACT Observed accretion rates onto the Milky Way and other local spirals fall short of that required to sustain star formation for cosmological timescales. A potential avenue for this unseen accretion is a rotating inflow in the volume-filling hot phase ($\sim 10^6\, {\rm K}$) of the circumgalactic medium (CGM), as suggested by some cosmological simulations. Using hydrodynamic simulations and a new analytic solution valid in the slow-rotation limit, we show that a hot inflow spins up as it approaches the galaxy, while remaining hot, subsonic, and quasi-spherical. Within the radius of angular momentum support ($\sim 15\, {\rm kpc}$ for the Milky Way) the hot flow flattens into a disc geometry and then cools from $\sim 10^6$ to $\sim 10^4\, {\rm K}$ at the disc–halo interface. Cooling affects all hot gas, rather than just a subset of individual gas clouds, implying that accretion via hot inflows does not rely on local thermal instability in contrast with ‘precipitation’ models for galaxy accretion. Prior to cooling and accretion the inflow completes ≈tcool/tff radians of rotation, where tcool/tff is the cooling time to free-fall time ratio in hot gas immediately outside the galaxy. The ratio tcool/tff may thus govern the development of turbulence and enhancement of magnetic fields in gas accreting onto low-redshift spirals. We show that if rotating hot inflows are common in Milky-Way-size disc galaxies, as predicted, then signatures of the expected hot gas rotation profile should be observable with X-ray telescopes and fast radio burst surveys.