  Given a Prym-Teichm\"uller curve in $\mathcal{M}_3$, this note provides an
invariant that sorts the cusp prototypes of Lanneau and Nguyen by component.
This can be seen as an analogue of McMullen's genus $2$ spin invariant,
although the source of this invariant is different. Moreover, we describe the
Galois action on the cusps of these Teichm\"uller curves, extending the results
of Bouw and M\"oller in genus $2$. We use this to show that the components of
the genus $3$ Prym-Teichm\"uller curves are homeomorphic.
