A knob direction and two dial numbers, and the two by two Hermitian matrix they produce, with its trace, determinant and eigenvalues.
knob tilt0°
knob swing0°
dial labels
eigenvalues 1, −1
Turn the knob. The entries rearrange. Trace, determinant and eigenvalues all hold still.
Retype a label. Trace, determinant and eigenvalues move. The knob has not shifted.
Where the four live. Two eigenvalues and two angles for the eigenbasis. The four matrix entries look like four numbers, but Hermiticity forces the diagonal real and pins the lower-left to the conjugate of the upper-right.
Identity. Set both labels equal and the knob stops mattering. A repeated eigenvalue leaves no eigenbasis to point.