Remington realized their mistake after a few years, and changed the rate of twist to the same as Winchester rifles, and re-named their rifles to "6 mm Remington".
The stuff I've read sez 6mm Rem rifles were initially offered w/ a 1:9, twist, while the .243 is generally 1:10 ...... the Original .244 Rem rifles used 1:12, IIRC.
Since your barrel is stamped .244 REM, it'd probably be the slower 1:12 .... but you could
just measure it yourself: you just need a .243/6mm bore brush on a cleaning rod and a sharpie ..... put the brush in the bore and mark 12:00 position back at the handle, and mark the rod at the reciever ring. Push the rod in until it makes one full revoloution (the rod will turn due to the rifling as it is pushed in....stop pushing when the mark by the handle gets back to 12:00 position ...... make another mark on the rod at the reciver ring. (Note: do NOT ever change directions with a bore brush while the brush is in the bore.) Push the brush on through the muzzle, remove. Withdraw the rods and measure the distance between the two marks- that would be 1 turn in X".