An interweaving relation is a Markovian similarity-type relation between two Markov chains introducing a warming-up time after which their time-marginal distributions can be tightly compared (for different initial distributions). For non-transient Markov transition kernels on the same state space, these relations are shown to be equivalent to the usual similarity relation. Some bounds are deduced on corresponding warming-up times, when the eigenvalues are furthermore assumed to be real. When the eigenvalues are non-negative, the same approach enables us to construct strong stationary times for irreducible Markov chains through interweaving relations with model absorbed Markov chains, thus extending a result due to Matthews in the reversible situation.