Ion Coulomb crystals (ICC) are formed by laser-cooled ions in both radio-frequency and Penning traps. In radio-frequency traps the crystals are generally stationary. In Penning traps ICC always rotate. The frequency of rotation is often set by an applied rotating wall drive which forces the crystal to rotate at the same frequency as the drive. In the absence of any applied rotating or oscillating fields, ICC in a Penning trap can be in stable equilibrium with a range of rotation frequencies. The density and shape of the crystal adjusts with the rotation frequency to ensure that equilibrium is reached. Here we show that the parameters of the radial laser cooling beam determine the rotation frequency of a small crystal in a Penning trap when no driving fields are present. We demonstrate, using an approximate theoretical treatment and realistic simulations, that the crystal rotation frequency is independent of the number of ions and the trap parameters, so long as the crystal radius remains smaller than the cooling laser beam waist. As the rotation frequency increases, the crystal eventually becomes a linear string, at which point it is no longer able to adjust its density. Instead, a small amplitude vibration in the zigzag mode of oscillation manifests itself as a rotation of the crystal at a fixed frequency that depends only on the applied trap potential.