The parameter identifiability problem for dynamic system ODE models has been extensively studied. Nevertheless, except for linear ODE models, the question of establishing identifiable combinations of parameters when the model is unidentifiable has not received as much attention and the problem is not fully resolved for nonlinear ODEs. Identifiable combinations are useful, for example, for the reparameterization of an unidentifiable ODE model into an identifiable one. We extend an existing algorithm for finding globally identifiable parameters of nonlinear ODE models to generate the 'simplest' globally identifiable parameter combinations using Gröbner Bases. We also provide sufficient conditions for the method to work, demonstrate our algorithm and find associated identifiable reparameterizations for several linear and nonlinear unidentifiable biomodels.