Categorical data occur in a wide range of statistical applications. If the data are observed in matched pairs, it is often of interest to examine the differences between the responses. The focus is on tests of axial symmetry in two-way tables. A commonly used procedure is the Bowker test which is a generalization of the McNemar test. The test decision is based on a chi-squared approximation which might not be adequate, for example if the table is sparse. Therefore, modifications of the test statistic have been discussed. A test of symmetry based on Bowker's test and Markov Chain Monte Carlo methods is proposed. A simulation study is carried out to determine and compare the performances of the simulation test, the Bowker test and two modifications.