Subspace identification can be used to obtain models of piecewise linear state-space systems for which the switching is known. The models should not switch faster than the block size of the Hankel matrices used. The nonconsecutive parts of the input and output data that correspond to one of the local linear systems can be used to obtain the system matrices of that system up to a linear state transformation. The linear systems obtained in this way cannot be combined directly, because the state transformation is different for each of the local linear systems. The transitions between the local linear systems can be used to transform the models to the same state space basis. We show that the necessary transformations can be obtained from the data, if the data contains a sufficiently large number of transitions for which the states at the transition are linearly independent. An algorithm to determine the transformations is presented, and the sensitivity with respect to noise is investigated using a Monte-Carlo simulation.