We analyse in detail the dynamics of a mechanical system which is a rigid body with the geometry of a double cone. This double cone is apparently able to spontaneously roll uphill along inclined rails. The experiment has been known for some centuries, and because of its peculiar behaviour, it has been named "mechanical paradox". Although this instrument is well known today, we have not found in the literature a dynamical study like the one we are presenting. A deeper analysis of this mechanical object will allow us to go further than explaining the apparent paradox in the system; it will show interesting features of the dynamics that are not evident or intuitive. In this work, we follow a complete study of the geometry, the kinematic variables and the Lagrangian dynamics of the problem for any set of the angular parameters and initial values, and obtain as a result a full description of the dynamic variables of this mechanical device. In addition to studying the dynamics of the system with the angles that yield the typical paradoxical behaviour, we study carefully what we call the constant potential geometry regime, where the centre of mass maintains its height, and found in this particular case some features of the dynamics which are not common in rolling objects. We believe that this work can offer the student good material to review some fundamental concepts of analytical mechanics. (Contains 16 figures.)