Vibrational sum-frequency generation spectroscopy (SFG) lineshapes of p-cyanobenzenethiol on low-index Ag crystal surfaces are studied as a function of azimuthal rotation by angle phi. A broadband multiplex SFG method is used, with a new technique that variably suppresses the non-resonant (NR) background using time-asymmetric time-delayed picosecond laser pulses. When both resonant (R) and NR signals are present, the amplitude and phase of the R line shape can vary significantly with phi, leading to dramatic phi-dependent variations of the SFG spectrum. The CN-stretch transition of p-cyanobenzenethiol modified Ag(111) and Ag(110) surfaces has an SFG spectrum consisting of a single vibrational resonance R atop a NR background that originates from the metal surface. Using the NR suppression technique, it was found that the R amplitude of the CN-stretch was independent of phi on both Ag(111) and Ag(110), which proves that the CN dipole moment is parallel to the surface normal in both cases. We show that it is possible to accurately extract the phi-dependence of the R amplitude, the NR amplitude, and the phase difference from SFG spectra by suppressing the NR signal during sample roation, thereby proving that the R contribution from the CN-stretch transition evidence phi-invariant behavior on both Ag surfaces.