From a broad perspective, this work deals with the question of reduction of filtration, i.e., given a stopping time θ relative to a full model filtration G, when and how to separate the information that comes from θ from a reference filtration in order to simplify the computations. Toward this aim, some kind of local martingale invariance property is required, but under minimal assumptions, so that the model stays as flexible as possible in view of applications (to, in particular, counterparty and credit risk). Specifically, we define an invariant time as a stopping time with respect to the full model filtration such that local martingales with respect to a smaller filtration and a possibly changed probability measure, once stopped right before that time, are local martingales with respect to the original model filtration and probability measure. The possibility to change the measure provides an additional degree of freedom with respect to other classes of random times such as Cox times or pseudo-stopping times that are commonly used to model default times. We provide an Azéma supermartingale characterization of invariant times and we characterize the positivity of the stochastic exponential involved in a tentative measure change. We study the avoidance properties of invariant times and their connections with pseudo-stopping times.