A Gaussian interference channel (IC) with a relay is considered. The relay is assumed to operate over an orthogonal band with respect to the underlying IC, and the overall system is referred to as IC with an out-of-band relay (IC-OBR). The system can be seen as operating over two parallel interferencelimited channels: The first is a standard Gaussian IC and the second is a Gaussian relay channel characterized by two sources and destinations communicating through the relay without direct links. We refer to the second parallel channel as OBR Channel (OBRC). The main aim of this work is to identify conditions under which optimal operation, in terms of the capacity region of the IC-OBR, entails either signal relaying and/or interference forwarding by the relay, with either a separable or non-separable use of the two parallel channels, IC and OBRC. Here "separable" refers to transmission of independent information over the two constituent channels. For a basic model in which the O...