the symmetric top
A symmetric top is a rigid body with an axis of symmetry, so that two of its three principal moments of inertia are equal (I_1 = I_2, and a distinct I_3 about the symmetry axis). A spinning toy top, a rifle bullet, a thrown football, a discus, and (to good approximation) the spinning Earth are all symmetric tops. The symmetry buys you an exactly solvable problem, which is why it is the standard model of spinning-body dynamics.
Two principal moments being equal makes the third axis special -- it is the figure axis, or symmetry axis. When I_3 < I_1 the body is prolate (cigar-shaped, like a bullet); when I_3 > I_1 it is oblate (disk-shaped, like a frisbee or the Earth). Because the inertia tensor has the form diag(I_1, I_1, I_3), rotation about the figure axis is torque-free, and Euler's equations decouple neatly: omega_3 is constant and omega_1, omega_2 rotate steadily. This is the case in which precession, nutation, and gyroscopic motion can all be worked out in closed form.
The symmetric top sits between the trivially simple spherical top (all three moments equal, e.g. a uniform sphere or cube) and the genuinely chaotic-prone asymmetric top (all three different, e.g. an arbitrary rock). It is the richest rotor that is still fully integrable, which is why every mechanics course, and much of molecular spectroscopy, uses it as the reference case.
A rifle barrel is 'rifled' with spiral grooves precisely to spin the bullet about its long (figure) axis. As a prolate symmetric top the bullet gains gyroscopic stability, keeping its nose pointed forward instead of tumbling -- the same physics as a spinning top staying upright.
A symmetric top has two equal principal moments, making its spin analytically solvable.
Do not confuse the shape symmetry with the inertia symmetry: what matters is I_1 = I_2, not that the object looks round. A square prism, though not round, is a symmetric top because its cross-sectional moments happen to be equal.