Our group first reported the operation of a relativistic backward wave oscillator (BWO) in the so-called `cross- excitation' regime in 1998. This instability, whose general properties were predicted earlier through numerical studies, was a consequence of using a particularly shallow rippled- wall waveguide (slow wave structure--SWS) that was installed in the experiment to diagnose pulse shortening in a long pulse electron beam-driven high power microwave (HPM) source. This particular SWS was required to accommodate laser interferometry measurements during the course of microwave generation. Since those early experiments we have further studied this regime in greater detail using two different SWS lengths. We have invoked time-frequency analysis, the smoothed-pseudo Wigner-Ville distribution in particular, to interpret the heterodyned signals of the radiated power measurements. These recent results are consistent with earlier theoretical predictions for the onset, voltage scaling, and general behavior for this instability. This paper presents data for a relativistic BWO operating in the single frequency regime for two axial modes, operating in the cross-excitation regime, and discusses the interpretation of the data, as well as the methodology used for its analysis. Although operation in the cross-excitation regime is typically avoided due to its poorer efficiency, we discuss how it may be exploited in HPM effects studies.