Although small (only 5mm long from head to tail) the flutter fly Palloptera muliebris is distinctive and easy to identify. Not only are their pretty tortoiseshell banded wings unmistakable, but their behaviour is odd enough to make you notice them.
Until recently it was assumed they came inside by accident, the way some hover flies do. So little is known for sure about their lifecycle that entomologists are just guessing, but there has been a suggestion that they may be predators of carpet beetles. This would mean the flies are coming inside deliberately, to lay their eggs so that their predatory larvae can hatch near a ready supply of food.
So, my advice if you see one of these flies in your house -- just let it get on with whatever it is doing (and check under your oriental rugs for woolly bears). The presence of a predator usually indicates a good population of prey.