The paper is concerned with one-dimensional two-sided Ornstein-Uhlenbeck type processes with delay or anticipation. We prove existence and uniqueness requiring almost sure boundedness on the left half-axis in case of delay and almost sure boundedness on the right half-axis in case of anticipation. For those stochastic processes (X, Pμ) we calculate the Radon-Nikodym density under time shift of trajectories, Pμ(dX·−t)/Pμ(dX), t 2 R.