Nearly all known extreme trans-Neptunian objects (ETNOs) have argument of perihelion close to 0 degrees. An existing observational bias strongly favours the detection of ETNOs with arguments of perihelion close to 0 degrees and 180 degrees yet no objects have been found at 180 degrees. No plausible explanation has been offered so far to account for this unusual pattern. Here, we study the dynamical evolution of comet 96P/Machholz 1, a bizarre near-Earth object (NEO) that may provide the key to explain the puzzling clustering of orbits around argument of perihelion close to 0 degrees recently found for the population of ETNOs. Comet 96P/Machholz 1 is currently locked in a Kozai resonance with Jupiter such that the value of its argument of perihelion is always close to 0 degrees at its shortest possible perihelion (highest eccentricity and lowest inclination) and about 180 degrees near its shortest aphelion (longest perihelion distance, lowest eccentricity and highest inclination). If this object is a dynamical analogue (albeit limited) of the known ETNOs, this implies that massive perturbers must keep them confined in orbital parameter space. Besides, its future dynamical evolution displays orbital flips when its eccentricity is excited to a high value and its orbit turns over by nearly 180 degrees, rolling over its major axis. This unusual behaviour, that is preserved when post-Newtonian terms are included in the numerical integrations, may also help understand the production of NEOs on retrograde orbits. ; Comment: 8 pages, 8 figures, 2 tables. Revised to reflect final version published in Monthly Notices of the Royal Astronomical Society