The following isn't as nice as using a driver, and it's not very precise but as long as you want the image sequence to start playing backwards from frame 1 of your animation, it kinda works:
Go to frame
1, keyframe the offset to the sequence's length minus one, then go to the frame where the sequence would have finished playing and keyframe the offset to the inverse of that.
For example, with an Image Sequence of 250 images:
1: Insert key for Offset = 249
250: Insert key for Offset = -249
To loop it, enable Cyclic, open a Graph Editor tab and add a Cycle Modifier to the Offset curve. Then go one frame past the last key (in my example: 251) and press I to add a key. Without that extra key the loop gets slowly out of sync.
Note: like I said, it's a bit rough, not frame-exact and sometime for some reason you might have to set the second offset to frame length - 2 for it to work. When using that trick in the compositor (second offset to 0) it's a lot more reliable and easier to work with the graph to change the speed.