We consider a model of two mutually delay-coupled semiconductor lasers (SLs) in a face to face configuration. The lasers are coherently coupled via their optical fields, where the time delay τ arises from the finite propagation time of the light from one laser to the other. This system is described well by single mode rate equations, which are a system of delay differential equations (DDEs) with one fixed delay. We study the compound laser modes (CLMs) of the system, where both lasers operate at an identical, time-independent frequency. By making use of numerical continuation applied to the full DDEs, we present a comprehensive geometrical picture of how CLMs depend on the two main physical parameters, namely, the coupling phase and the detuning between the two lasers. The different branches of CLMs are organized by unfoldings of pitchfork bifurcations that exist for zero detuning. As a function of the detuning, different branches of CLMs connect, split, or disappear in transitions through codimension-one singularities in the surface of CLMs. © 2006 Society for Industrial and Applied Mathematics.