The increase of information volumes, which are transmitted in modern satellite telecommunication systems, requires the development of new signal processing technologies, microwave devices, antenna systems and methods of their analysis. In particular, polarization-adaptive antennas are widely used for this purpose. Such antennas provide the possibility to transmit and receive radio signals with polarization of any type. Polarization processing devices of antenna systems must provide low levels of voltage standing wave ratio of waves with horizontal and vertical linear polarizations and high crosspolar discrimination simultaneously. Therefore, there is the need to improve designs and methods of analysis of modern polarization processing devices. Polarizers based on a square waveguides with irises are widely used due to the simplicity of their design and manufacturing by milling technology. The article considers a new mathematical model of waveguide polarizers with reactive irises. For the example of model application we have simulated and optimized a polarizer based on a square waveguide with four irises. A mathematical model of this waveguide polarizer was developed based on the description of microwave devices and their elements by wave scattering and transmission matrices. The general scattering matrix of the polarizer has been obtained analytically. The main electromagnetic characteristics of the polarizer were determined based on the elements of this matrix. As a result, we have analyzed main characteristics of the model, including differential phase shift, voltage standing wave ratio for vertical and horizontal polarizations, axial ratio and crosspolar discrimination. The optimization of the characteristics of a polarizer has been performed using the developed mathematical model and a software based on the finite integration technique. The optimal characteristics and geometrical sizes of the structure are in good agreement, which proves the correctness of the developed mathematical model of square waveguide iris polarizers.