Context. Rotational mixing transports angular momentum and chemical elements in stellar radiative zones. It is one of the key processes for modern stellar evolution. In the past two decades, an emphasis has been placed on the turbulent transport induced by the vertical shear instability. However, instabilities arising from horizontal shear and the strength of the anisotropic turbulent transport that they may trigger remain relatively unexplored. The weakest point of this hydrodynamical theory of rotational mixing is the assumption that anisotropic turbulent transport is stronger in horizontal directions than in the vertical one.Aims. This paper investigates the combined effects of stable stratification, rotation, and thermal diffusion on the horizontal shear instabilities that are obtained and discussed in the context of stellar radiative zones.Methods. The eigenvalue problem describing linear instabilities of a flow with a hyperbolic-tangent horizontal shear profile was solved numerically for a wide range of parameters. When possible, the Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) approximation was applied to provide analytical asymptotic dispersion relations in both the nondiffusive and highly diffusive limits. As a first step, we consider a polar f-plane where the gravity and rotation vector are aligned.Results. Two types of instabilities are identified: the inflectional and inertial instabilities. The inflectional instability that arises from the inflection point (i.e., the zero second derivative of the shear flow) is the most unstable when at a zero vertical wavenumber and a finite wavenumber in the streamwise direction along the imposed-flow direction. While the maximum two-dimensional growth rate is independent of the stratification, rotation rate, and thermal diffusivity, the three-dimensional inflectional instability is destabilized by stable stratification, while it is stabilized by thermal diffusion. The inertial instability is rotationally driven, and a WKBJ analysis reveals that its growth rate reaches the maximum value of √f(1 − f) in the inviscid limit as the vertical wavenumber goes to infinity, where f is the dimensionless Coriolis parameter. The inertial instability for a finite vertical wavenumber is stabilized as the stratification increases, whereas it is destabilized by the thermal diffusion. Furthermore, we found a selfsimilarity in both the inflectional and inertial instabilities based on the rescaled parameter PeN2 with the Péclet number Pe and the Brunt–Väisälä frequency N.