When Gigolo is in the system tray and you turn on or off compositing, the window will open up. Gigolo 0.5.1 / Xubuntu 20.04
This happens because the systray area is recreated and that forces gigolo to display its main window. https://gitlab.xfce.org/xfce/xfce4-panel/-/blob/master/plugins/systray/systray.c#L444 https://gitlab.xfce.org/apps/gigolo/-/blob/master/src/window.c#L1197
Yes, that's accurate (so not a bug).
Strange that this would be classified as not a bug when other apps don't show this same behavior, but whatever.
FWIW, the compositor is not meant to be started and stopped every so often.