Recently the need for processing three-dimensional (3-D) data has arisen in geophysics and computer tomography. In the paper the fundamental symmetries of 3-D digital IIR (infinite impulse response) filters are described and some theorems which give necessary and sufficient conditions that a filter exhibits specifical symmetries are given. The filter is given as the ratio of two polynomials in \(z_ 1\), \(z_ 2\) and \(z_ 3\). Finally, it is shown how the occurrence of these symmetries lead to a reduction of computation time when one uses computer-aided design. Some examples are presented.