the principal axes of inertia
The principal axes are the three special, mutually perpendicular directions in a body for which spinning is 'clean': spin the body about a principal axis and its angular momentum lines up exactly with the axis, with no wobble. Every rigid body has such a natural set of axes, usually aligned with its symmetries -- the long axis of a rod, the symmetry axis of a cone, the three edges of a brick.
Mathematically they arise because the inertia tensor is a real symmetric matrix, so it can be diagonalized by an orthogonal (rotation) transformation. The principal axes are its eigenvectors, and the corresponding eigenvalues I_1, I_2, I_3 are the principal moments of inertia, found from det(I - lambda 1) = 0. In the frame aligned with these axes the inertia tensor is diagonal, all products of inertia vanish, and the physics becomes simple: L_1 = I_1 omega_1, L_2 = I_2 omega_2, L_3 = I_3 omega_3 componentwise, and T = (1/2)(I_1 omega_1^2 + I_2 omega_2^2 + I_3 omega_3^2).
This is why almost all rigid-body dynamics is done in the principal-axis body frame: it turns a coupled tangle into three clean equations, and it is the natural frame for Euler's equations. Practically, the principal axes are the axes about which a body can spin without vibrating -- which is exactly what a wheel-balancing machine finds when it removes the products of inertia that make a car wheel shudder.
Throw a book spun about its shortest axis or its longest axis and it tumbles stably; spin it about the intermediate axis and it flips chaotically end over end. All three are principal axes, but the intermediate one is unstable -- the famous tennis-racket (Dzhanibekov) effect.
Rotation about the axes of largest and smallest inertia is stable; about the intermediate axis it is not.
Principal axes always exist and are orthogonal because the inertia tensor is symmetric -- this is the spectral theorem, not luck. When two principal moments are equal the axes in that plane are not unique (any perpendicular pair will do), which is exactly the case of the symmetric top.