In the present paper we demonstrate that the iron-mineral-based magnetoreceptor model can providebirds with a magnetic compass in addition to the generally believed "magnetic map". We show that the iron-mineral-based magnetoreceptor system possesses all properties of a polarity compass, which is extremely important for avian navigation. We study how parameters of the magnetoreceptor system influence on the properties of the compass and show that at certain conditions it acquires features of an inclination compass. In the present paper we address the question of avian magnetoreception theoretically and suggest several experiments which should be performed for better understanding of the iron-mineral-based compass in birds.