Saturn jumping

Hi I am a woman but I wont make excuses, because I know a great deal about cars but this one has me very confused! I own a 97 Saturn SL1 when I bought it it had 125, 000 miles. It now has 142, 000 miles on it. Ever since I bought it the engine jumps. Now here is the tricky part, when I went to get an oil change I told the mechanic about it. I told him that it ONLY jumps when I am putting it in reverse and ONLY when the car is warmed up. Those two things togwther. In the morning when I go to back out of the garage, or after work, once the car has cooled down, it NEVER jumps. Once the engine warms up everytime there after it will hesitate and then jump into reverse. It never does this putting it in drive ONLY reverse. Now I have asked every man and every mechanic I know, and I let someone else put it in reverse and looked with the hood up, and I could see the whole engine shake and move backwards when he put it in reverse. So the other mechanic standing with me said that my engine mounts needed to be replaced, others have said transmission, then I have heard change the trans fluid, then I heard valve pack, valve body replacement. CAN SOMEONE WHO KNOWS WHAT EXACTLY IS THE PROBLEM PLEASE TELL ME HONESTLY! The car runs great and this is the only issue, I just want to get it fixed, look how many miles and almost 3 yrs I have been driving like this. I think if it was the trans it would have went out by now and I think drive would also be effected. Help me somebody! Thank you so very much.
LISA