A finite mixture model is introduced for the unsupervised classification of three-way ordinal data. Specifically, the finite mixture of Gaussians is observed by a discretized version of its variables. The approach focuses on reducing the number of model parameters by identifying a subspace that contains the information sufficient to classify the observations. This process also helps to detect noise variables and/or occasions. The group-specific means and covariances are reparameterised using parsimonious models taking into account the three-way structure of the data. Estimation is performed using a composite likelihood approach to reduce the computational complexity. Parameter estimates are computed by means of an EM-like algorithm.