Due to their indeterminacies, static and dynamic factor models require identifying assumptions to guarantee uniqueness of the parameter estimates. The indeterminacy of the parameter estimates with respect to orthogonal transformations is known as the rotation problem. The typical strategy in Bayesian factor analysis to solve the rotation problem is to introduce ex-ante constraints on certain model parameters via degenerate and truncated prior distributions. This strategy, however, results in posterior distributions whose shapes depend on the ordering of the variables in the data set. We propose an alternative approach where the rotation problem is solved ex-post using Procrustean postprocessing. The resulting order invariance of the posterior estimates is illustrated in a simulation study and an empirical application using a well-known data set containing 120 macroeconomic time series. Favorable properties of the ex-post approach with respect to convergence, statistical and numerical accuracy are revealed.