A meter and some measurements would quickly tell you what's going on
You didn't specify what inverter
But you did say inverter charger
It is normal for them to be turned on
If it is all the way off, you won't get any charging, when on shore power
Just because you turned it on does not mean it is inverting
There will be shore power pass thru
But the unit may have to be on before the pass thru works
If it is in charge mode while on shore power, it is not inverting and using battery, if it was not on it would not be charging
When in doubt read the instructions
Posting the make and model here will get you more info