ABSTRACT Gravitational waves from binary black hole mergers have allowed us to directly observe stellar-mass black hole binaries for the first time and therefore explore their formation channels. One of the ways to infer how a binary system is assembled is by measuring the system’s orbital eccentricity. Current methods of parameter estimation do not include all physical effects of eccentric systems, such as spin-induced precession, higher order modes, and the initial argument of periapsis: an angle describing the orientation of the orbital ellipse. We explore how varying the argument of periapsis changes gravitational waveforms and study its effect on the inference of astrophysical parameters. We use the eccentric spin-aligned waveforms TEOBResumS and SEOBNRE to measure the change in the waveforms as the argument of periapsis is changed. We find that the argument of periapsis could already be impacting analyses performed with TEOBResumS. However, it is likely to be well resolvable in the foreseeable future only for the loudest events observed by LIGO–Virgo–KAGRA. The systematic error in previous, low-eccentricity analyses that have not considered the argument of periapsis is likely to be small.