Relay control is one of the simplest methods for stabilizing open-loop unstable processes by producing limit cycles. It is also used for parameter identification of unstable processes for the design of other types of controllers. In this paper, exact analysis of the limit cycles has been obtained for a class of high-order unstable SISO processes under relay control. A key technical lemma provides two nonlinear equations whose solutions provide the time periods of the upswing and downswing modes. Analytical and graphical methods are used to determine the existence and multiplicity of the limit cycles. Also, necessary and sufficient conditions have been developed to determine the stability of the limit cycles. These tools then allow for the bifurcation analysis of the limit cycles based on variations in time delay. One interesting result is the presence of irregular stabilization, where increasing time delay could regain stability that was lost at smaller time delays. The results are then combined to provide a set of necessary conditions for relay stabilization. These conditions can be represented by a compact pyramid region which then yields some useful guidelines for the synthesis of additional compensators for relay-stabilization.