You may have a dual fuel system with a furnace and a heat pump. Do you know what fuel is supplied to the furnace?

In a dual fuel system if the heat pump is being used, the furnace blower would come on but it would not fire. If it did it could over heat the heat exchanger and damage it.

On very cold days, the heat pump would not operate. The furnace would fire and heat the house.

If you have concerns it would be best to have an HVAC professional do a thorough inspection.

You need to figure out whether you have a straight cool air conditioner or a heat pump. Maybe what you think is the air conditioner turning on is actually the heat pump. Is the indoor unit an electric air handler, or a gas furnace?

If you have a straight cool unit, then no this is not normal. Probably a wire crossed somewhere.

It is a split system with a gas furnace inside and the AC condensing unit outside. I am leaning toward the crossed wire like you suggested but the home inspector mentioned the possibility of "Hybrid" which I am not at all familiar with. Just thought I would throw out the question and see what came from it. Thank you.

OK, if the indoor unit is gas, it should not be firing at the same time as the outdoor unit as Mike said. The one exception would be if it is a hybrid system and the heat pump is defrosting, but I highly doubt that would be the case since it happened right when you fired it up. There is probably something wrong the seller will need to get fixed.

When was this AC condenser installed? Operating an AC in the winter may cause damage to the compressor. It is hard to believe this has been cross wired for a long time, but again anything is possible.

Are you paying for this kind of nonsense? The inspecter owes you an accurate opinion on all systems,not just those he understands. IMO,it's up to the inspection company to send out someone else to answer these questions.