Before revision:  While this underlying involution proposal structure suggests a scope which includes Hamiltonian-type kernel structures , we demonstrate that our result also recovers a broad range of kernels encompassed by a previous general state space approach in [Tierney, Annals of Applied Probability, 1998]  . Altogether, the theoretical unity and reach of our main result provides a basis for deriving novel sampling algorithms while laying bare important relationships between existing methods.
After revision:  While this underlying involution proposal structure suggests a scope which includes Hamiltonian-type kernel structures , we demonstrate that our result also recovers a broad range of kernels encompassed by a previous general state space approach in [Tierney, Annals of Applied Probability, 1998] where the connection to Hamiltonian methods was more obscure . Altogether, the theoretical unity and reach of our main result provides a basis for deriving novel sampling algorithms while laying bare important relationships between existing methods.