We study orthogonal projections of embedded surfaces M in H3+ (−1) along horocycles to planes. The singularities of the projections capture the extrinsic geometry of M related to the lightcone Gauss map. We give geometric characterizations of these singularities and prove a Koenderink-type theorem that relates the hyperbolic curvature of the surface to the curvature of the profile and of the normal section of the surface. We also prove duality results concerning the bifurcation set of the family of projections.