We investigate flow instability produced by viscosity and density discontinuities at the interface separating two Newtonian fluids in generalised Couette–Poiseuille (GCP) flow. The base flow, driven by counter-moving plates and an inclined pressure gradient at angle $0^\circ \leqslant \phi \leqslant 90^\circ$, exhibits a twisted, two-component velocity profile across the layers, characterised by the Couette–Poiseuille magnitude parameter $0^\circ \leqslant \theta \leqslant 90^\circ$. Plane Couette–Poiseuille (PCP) flow at $ \phi = 0^\circ$ is considered as a special case. Flow/geometry parameters are $(\phi ,\theta )$, a Reynolds number $Re$ and the viscosity, depth and density ratios $(m,n,r)$, respectively. A mapping from the GCP to PCP extended Orr–Sommerfeld equations is found that simplifies the numerical study of interfacial-mode instabilities, including determination of shear-mode critical parameters. For interfacial modes, unstable regions in $(m,n,r)$ space are delineated by three distinct surfaces found via long-wave analysis, with the exception of strict Couette flow where the $(m,n)$ surface asymptotically vanishes with $\theta \rightarrow 0^\circ$. In interfacial stable regions but with unstable shear modes, one-layer PCP stability can be identified with a cut-off $\theta$ that conforms to canonical PCP stability. Competition between the interfacial-mode reversal phenomenon and the shear-mode cut-off behaviour is discussed. Extending to the full GCP configuration with the mapping algorithms applied, we systematically chart how pressure-gradient inclination and perturbation wavefront angle shift the balance between interfacial and shear instabilities in a specific case.