The paradoxical mechanism of P. L. Chebyshev is the main issue of this article. The paradoxical mechanism could be considered as the union of the lambda-mechanism and a double pendulumthe free vertices of which are hingedly connected. One of the pendulum rods is usually replaced by a massive disk. Among all known mechanisms of P. L. Chebyshev, singular points of the configuration space or branching points arise only in the paradoxical mechanism. In particular configurations of the paradoxical mechanism the rods of the double pendulum become parallel. In this case, the mechanism could continue the motion from the singular points in two different ways. There are two types of smooth motion of the paradoxical mechanism, which differ by the direction of the disk rotation. In the first case, with a full circle of the drive link, the disk makes two full circles. In the second case, it makes four full circles. The trajectory of the free vertex of the lambda-mechanism in the paradoxical mechanism is located between two concentric circles and alternately touches each circle at three points. In addition, the curve that is located between two concentric circles and touches each circle in an arbitrary number of points is considered in the article. It is proved that it is possible to obtain any given number of disk revolutions in one revolution of the mechanism-driving link. The basic equations of dynamics are written down for the paradoxical mechanism. The expression for the moment of inertia forces of the disk is written down. The work of the external moment, which is applied to the leading link of the mechanism, and the work of the disk inertia forces are compared.