Steady methane/air laminar premixed flames stabilised on a cylindrical bluff body subjected to a continuous rotation are analysed using joint direct numerical simulations (DNS) and experiments. DNS are carried out using a 19 species scheme for methane/air combustion and a lumped model to predict the cylinder temperature. Rotation of the cylinder induces a symmetry breaking of the flow, and leads to two distinct flame branches in the wake of the cylinder. DNS are validated against experiments in terms of flame topologies and velocity fields. DNS are then used to analyse flame structures and thermal effects. The location and structure of the two flames are differently modified by rotation and heat transfer: a superadiabatic flame branch stabilises close to the hot cylinder and burns preheated fresh gases while a subadiabatic branch is quenched over a large zone and anchors far downstream of the cylinder. Local flame structures are shown to be controlled to first order by the local enthalpy defect or excess due to heat transfer between the cylinder and the flow. An analysis of the local wall heat flux around the cylinder shows that, for low rotation speeds, the superadiabatic flame branch contributes to wall heat fluxes that considerably exceed typical values found for classical flame/wall interactions. However, for high rotation speeds, fluxes decrease because the cylinder is surrounded by a layer of burned gases that dilute incoming reactants and shield it from the flame.