Markov switching autoregressive models (MSARMs) are efficient tools to analyse nonlinear and non-normal time series. A special MSARM with a hidden state-dependent seasonal component is proposed here to analyse periodic time series. We present a complete Metropolis-within-Gibbs algorithm for constraint identification, for model choice and for the estimation of the unknown parameters and the latent data. These three consecutive steps are developed tackling the problem of the hidden states labeling, by means of random permutation sampling and constrained permutation sampling. The missing observations occurring within the observed series and the future values are respectively estimated and forecasted considering them as unknown parameters. We illustrate our methodology with an example about the dynamics of an air pollutant.