We will describe a new interpretation of bordered Heegaard Floer invariants in the case of a manifold M with torus boundary. In our setting, these invariants, originally defined as homotopy classes of (differential) modules over a particular algebra, take the form of homotopy classes of immersed curves in the boundary of M decorated with local systems. Moreover, pairing two bordered Floer invariants corresponds to taking the Floer homology of two sets of decorated immersed curves. In most cases this simply counts the minimal intersection number, which leads to a number of applications. This is joint work with Liam Watson and Jake Rasmussen.