In this paper, we study invariant control systems that generalise positive systems. A characterisation of linear control systems invariant on polyhedral cones (corner regions) in the state-space, called cone-invariant linear control systems, is established both for the inputs taking values in a polyhedral cone in the control space and for the inputs taking values in an affine polyhedral cone. The problem of equivalence between control systems invariant on corner regions is introduced. For cone-invariant linear control systems, we study invariance-preserving state-equivalence and invariance-preserving feedback-equivalence and present characterisations of both notions of equivalence.