A bit of clarification, Paul.
Do you have a User's Manual? If you haven't picked one up, please do so. You can also get it in a pdf from the Honda owner's site. The Owner's Guide is good, but it is not as comprehensive. Page 12 of the User's Manual has a chart indicating when the Engine kicks in when you are in EV mode, depending on which one of the various driving modes: ECON, NORMAL, or SPORT you have selected. It depends on how much power you are requesting from the battery, which may request added boost from the Engine. The three driving modes create different mappings of the accelerator pedal, so the same press of the pedal in SPORT mode will request more acceleration than identical press in NORMAL, than in ECON. SPORT will thus appear more "lively", the ECON will make your driving appear more "smooth".
In HV mode, the power requests of the pedal are still determined by what driving mode you are in and may result in supplementary battery use as well as engine use. So it is always good to have some charge available for the system.
Page 11 of the User's Manual describes all possibilities. Unfortunately it does not specify exactly what specific states of the system result in changes of the topology of the energy flow of the running system as you drive.
I learned a lot by driving with the energy flow screen visible on the Information Screen. Recommend it!