PA-24 Comanche pilot reported an engine failure on takeoff and immediately landed on remaining runway; coasting to a stop past the runway hold lines on the taxiway.

Narrative

I received clearance for and initiated an intersection takeoff from Runway XXL at [taxiway] 1 at the ZZZ airport. The aircraft accelerated smoothly; and engine instrumentation showed normal readings before rotation. After rotation and at about 120 feet AGL; I moved the gear to the UP position; simultaneously heard a loud sound I would describe as metal breaking; and felt the aircraft slow. I recognized this as an engine failure. I moved the gear selector back to the down position and quickly evaluated my options - land on the remaining runway with a chance of a runway excursion on roll-out or glide past the airport perimeter and set down in trees. I elected to land on the remaining runway. From the moment I heard the noise; all training took over; and I only specifically remember doing two things. First; ensure the gear went back down; and second; make a brief radio call '<call sign> - request priority.' There was no time to troubleshoot; run a checklist; or describe my situation to a tower controller. I landed just before the X turnoff; applied maximum braking; and turned the aircraft to exit on taxiway 2 - I coasted to a stop past the hold short lines. During the roll-out; the propeller stopped rotating. The aircraft did not sustain any damage; and I did not sustain any injuries. After the aircraft was towed to a parking position; I sumped fuel from the tank used for takeoff - the fuel was clean and dry. I then ran my 'Aircraft Shutdown & Parking' checklists. During this process; I moved the mixture to idle-cut-off; and the mixture handle kept moving - until I could pull past the detent and pull the cable into the cockpit - the mixture cable had broken or come loose. I spoke with the manufacturer of my carburetor - the mixture should not default to 'cutoff' if a cable breaks; rather; it should stay in the same position. The noise I heard may indicate that the cable broke under some kind of tension; which turned the mixture to 'cut off.' My mechanic won't be able to look at the aircraft for another week - I will file another report if pertinent information comes to light on the mechanical condition. Of note; aircraft records indicate that the mixture cable is original to the aircraft. I was asked by another pilot why I started gear retraction with the runway remaining. I have a simple answer - my perspective of what I am willing to land on in an emergency has radically changed. The plane is undamaged; and I am unhurt - in the same circumstances; I'd hope I would take the same actions. However; I recognize that I didn't review my 'engine out on take-off and climb' briefing - which I would have done if there was another pilot in the aircraft with me. It may have better mentally prepared me for this event.