We study the relationship between certain SLE$_κ(ρ)$ processes, which are variants of the Schramm-Loewner evolution with parameter $κ$ in which one keeps track of an extra marked point, and Liouville quantum gravity (LQG). These processes are defined whenever $ρ> -2-κ/2$ and in this work we will focus on the light cone regime, meaning that $κ\in (0,4)$ and $\max(κ/2-4,-2-κ/2) < ρ< -2$. Such processes are self-intersecting even though ordinary SLE$_κ$ curves are simple for $κ\in (0,4)$. We show that such a process drawn on top of an independent $\sqrtκ$-LQG surface called a weight $(ρ+4)$-quantum wedge can be represented as a gluing of a pair of trees which are described by the two coordinate functions of a correlated $α$-stable Lévy process with $α= 1-2(ρ+2)/κ$. Combined with another work, this shows that bipolar oriented random planar maps with large faces can be identified in the scaling limit with an SLE$_κ(κ-4)$ curve on an independent $\sqrtκ$-LQG surface for $κ\in (4/3,2)$.