Let $h$ be an instance of the GFF. Fix $\kappa \in (0,4)$ and $\chi =2/\kappa -\kappa /2$. Recall that an {\em imaginary geometry ray} is a flow line of $ei(h/\chi +\theta )$ that looks locally like $\backslash SLE\kappa$. The {\em light cone} with parameter $\theta \in [0,\pi ]$ is the set of points reachable from the origin by a sequence of rays with angles in $[-\theta /2,\theta /2]$. It is known that when $\theta =0$, the light cone looks like $\backslash SLE\kappa$, and when $\theta =\pi$ it looks like the range of an $\backslash SLE16/\kappa$ {\em counterflow line}. We find that when $\theta \in (0,\pi )$ the light cones are either fractal carpets with a dense set of holes or space-filling regions with no holes. We show that every non-space-filling light cone agrees in law with the range of an $\backslash SLE\kappa (\rho )$ process with $\rho \in ((-2-\kappa /2)\vee (\kappa /2-4),-2)$. Conversely, the range of any such $\backslash SLE\kappa (\rho )$ process agrees in law with a non-space-filling light cone. As a consequence of our analysis, we obtain the first proof that these $\backslash SLE\kappa (\rho )$ processes are a.s.\ continuous curves and show that they can be constructed as natural path-valued functions of the GFF.