The influence of toroidal plasma rotation on the existence of reversed shear Alfvén eigenmodes (RSAEs) near their minimum frequency is investigated analytically. An existence condition is derived showing that a radially decreasing kinetic energy density is unfavourable for the existence of RSAEs. The Coriolis effect is typically unfavourable for modes moving slower than the plasma or moving in the opposite direction. The generality of the analysis allows for the description of downwards sweeping RSAEs and also rotation-induced modes in regular shear plasmas.