In conformal differential geometry, there are some distinguished curves, often known as “conformal circles,” since, on the round sphere, they are the round circles (and these are conformally invariant). But on the two-sphere, the curves of constant compass bearing are also conformally invariant. These “loxodromes” admit a curved analogue in the realm of Möbius geometry. In this article, these curved analogues are explained and the fifth-order invariant ODE that they satisfy is derived.