A head-mounted device may have displays that provide images. Waveguides may be used in conveying the images to eye boxes. The waveguides may overlap lenses in a glasses frame or other head-mounted support structure. The waveguides and lenses may be transparent. This allows real-world objects to be viewed from the eye boxes. Infrared-light reflectors may overlap the lenses. Gaze tracking system light sources may supply infrared light that reflects from the infrared-light reflectors to the eye boxes to illuminate a user's eyes. Gaze tracking system cameras capture gaze tracking images of the eyes from the eye boxes to track the user's gaze. Fiducials associated with the infrared-light reflectors may be monitored using the gaze tracking system cameras. This allows components such as the gaze tracking system cameras to be calibrated.